Washington (CNN) Our weekly roundup of the news, notes and chatter about the prospects for the next Democratic presidential race:

Add another major 2020 Democratic player to the list of supporters of single-payer health insurance: Sen. Kirsten Gillibrand.

"Yes," the New York senator does support single-payer, her senior adviser Glen Caplin told me.

Gillibrand first seemed to endorse the idea on the steps outside the Capitol this week, in a Facebook Live hosted by New Jersey Sen.. "Health care should be a right, it should never be a privilege. We should have Medicare for all in this country," she said.

However, Gillibrand in the past hasn't used the phrase "Medicare for all" as a substitute for "single-payer" the same way Bernie Sanders does. Instead, as Caplin pointed out, "since she first ran for Congress in 2006 in a red district, Kirsten has been advocating for 'Medicare for all' where anyone can buy into Medicare for a price they can afford" — that is, by paying a fixed percentage of their income.

Elizabeth Warren, who That all begged the follow-up: Does Gillibrand support single-payer? Her senior adviser's answer — "yes" — to that question is a major development for Gillibrand. It positions her with Sanders, progressive activists, and as of this week, Massachusetts Sen., who told The Wall Street Journal that on health care, "now it's time for the next step. And the next step is single-payer." Per the Pew Research Center , 52% of Democrats support single-payer.

Democrats generally want to set aside the topic of single-payer while they are fighting President Donald Trump's effort to repeal and replace the Affordable Care Act, also known as Obamacare. But for the 2020 Democratic presidential race, embracing it increasingly seems to be the price of admission into the upper tier.

What we're watching: Which Democratic senators will sign on as cosponsors when Sanders filed his single-payer bill? That bill is expected to drop after the Senate Republican push to repeal and replace President Barack Obama's signature domestic legislation reaches its conclusion.

News and notes:

Cory Booker John Lewis of Georgia, the civil rights icon. But more and more Democrats joined them outside the Capitol, and it wound up going more than three hours. The episode was a reminder that Booker — a pioneer on political Twitter as Newark mayor — has a magnetic personality that shines through on social media in a way other 2020 prospects might struggle to match. Booker's Facebook Live: That Facebook Live started with just the New Jersey senator and Rep.of Georgia, the civil rights icon. But more and more Democrats joined them outside the Capitol, and it wound up going more than three hours. The episode was a reminder that Booker — a pioneer on political Twitter as Newark mayor — has a magnetic personality that shines through on social media in a way other 2020 prospects might struggle to match.

— Booker's ties to Wall Street are too close for many progressives' comfort — and sometimes, as in his refusal to join most Democrats in calling for a full repeal of Jared Kushner's security clearance, he plays it safe. But as one Democratic operative put it: "I don't think litmus tests are going to matter as much as energy and authenticity and Booker has both. ... People find him personally compelling. You can't beat that with a policy critique and that was proven out again on this livestream this week."

Jeff Sessions on criminal sentencing for drug users, accusing Sessions of " — Booker also wrote a letter to the editor of The Washington Post rebutting Attorney Generalon criminal sentencing for drug users, accusing Sessions of " using the politics of fear ."

The Senate's "no" caucus: Booker, Gillibrand and others, including Minnesota Sen. Al Franken, Virginia Sen. Tim Kaine and Connecticut Sen. Chris Murphy, have said they don't want to run for president. OK ... but should we believe them? I asked some smart Democrats, and most said they really do believe Franken won't run. As for the rest — and any other top-level Democrat — it's particularly dangerous at this stage to be seen as fighting for yourself, rather than against Trump.

Joe Biden visited a Delaware pool Monday where years ago he was the only white lifeguard as a child. The pool is being renamed after him — and Biden also used the visit to make a statement about race and poverty. He said when he worked there, the other lifeguards would "ask me questions — because I really was the only white guy they really knew — about things that just startled me," Biden said. "I remember one lifeguard asked if I had a five-gallon can for gasoline. I said 'No, I don't. But what do you need it for?' He said, 'I'm going down to see my grandmom in North Carolina. We can't stop at most gas stations. Biden the lifeguard: Former Vice Presidentvisited a Delaware pool Monday where years ago he was the only white lifeguard as a child. The pool is being renamed after him — and Biden also used the visit to make a statement about race and poverty. He said when he worked there, the other lifeguards would "ask me questions — because I really was the only white guy they really knew — about things that just startled me," Biden said. "I remember one lifeguard asked if I had a five-gallon can for gasoline. I said 'No, I don't. But what do you need it for?' He said, 'I'm going down to see my grandmom in North Carolina. We can't stop at most gas stations. They won't let us stop at most gas stations .' I learned a lot."

Joe and Jill Biden also andalso launched Monday the new Biden Cancer Initiative, a private venture meant to work in tandem with Biden's "Moonshot" program.

Elizabeth Warren Samantha Bee this week, telling her that the way to respond to Trump's presidency is to fight. "It means that when you're told to be quiet, you don't sit down and be quiet," Warren said. "You stand back up." The line shows Warren is still leaning on Senate Majority Leader Mitch McConnell's Warren's advice: Massachusetts Sen. gave a bit of a pep talk tothis week, telling her that the way to respond to Trump's presidency is to fight. "It means that when you're told to be quiet, you don't sit down and be quiet," Warren said. "You stand back up." The line shows Warren is still leaning on Senate Majority Leader's admonishment earlier this year.

— Warren directly courted CEOs on health care in a Forbes op-ed , writing that "a cold look at the bottom line should be enough to make the business case for preserving the gains achieved by the Affordable Care Act."

— There's a new super PAC called "Deal Her Out" focused on defeating Warren in her 2018 re-election bid.

Chris Murphy got his first Dominic Rapini, a Branford businessman who was inspired by Trump's win. Murphy gets a challenger: Connecticut Sen.got his first challenger headed into the 2018 midterms:, a Branford businessman who was inspired by Trump's win.

Mark Zuckerberg, Oprah Winfrey, Mark Cuban, Starbucks founder Howard Schultz, Dwayne "The Rock" Johnson and actor Tom Hanks among the Democratic 2020 prospects, extending an invitation to meet with its editorial board to all of them. (We've heard Schultz is the one to watch in that group.) The "stars" of 2020: Speaking of New Hampshire, the Concord Monitor named Facebook's, Starbucks founderand actoramong the Democratic 2020 prospects, extending an invitation to meet with its editorial board to all of them. (We've heard Schultz is the one to watch in that group.)

The week ahead:

— Monday, July 3: California Sen. Kamala Harris will speak at a citizenship ceremony for 41 children and young people from 14 countries. They will be sworn in as US citizens aboard the USS Iowa in San Pedro at 9:15 a.m. PT. She'll also attend a health care rally an hour later in Torrance organized by the SEIU and the #Fight4OurHealth coalition, and then a roundtable on health care at 2:45 p.m. PT in San Francisco.

Jason Kander's visit to Iowa. On July 6, he's — Thursday, July 6: We talked last week about's visit to Iowa. On July 6, he's headed to New Hampshire , where he'll deliver Gov. Chris Sununu a petition opposing a Republican bill that tightens voting and registration requirements there.

Ben Sasse, a prominent Trump critic, will Jeff Kaufmann recently lambasted Sasse for his Trump criticism, — Friday, July 7: A Republican event to watch: Nebraska Sen., a prominent Trump critic, will speak at a Story County, Iowa, GOP dinner at 6 p.m. CT. Iowa Republican chairmanrecently lambasted Sasse for his Trump criticism, saying : "You know what Sen. Sasse, I really don't care what you like — we love Donald Trump! And if you don't love him, I'd suggest you stay on your side of the Missouri River."

The view from the left:

Notes from CNN's Greg Krieg:

Former Ohio state Sen. Nina Turner, one of Bernie Sanders' top surrogates during the 2016 primary contest, is the new president of "Our Revolution," the political organization spawned by his campaign. She replaces former Sanders campaign manager Jeff Weaver.

Why does it matter? Our Revolution owns the vaunted Sanders email list. But its influence — so far — has not matched its founding promise. If it's going to be a player in 2020, it needs to back (and attract) winners in 2018. Turner's new job is a nod to that. As board chair Larry Cohen told me in a text message last night, "Our strategy is very much based on building the membership in our hundreds of groups that focus on issues, candidates and party building and the connections between the three. This is much harder work than simply fighting Trump."

Translation: Turner, an electric speaker and beloved on the progressive left, can fire up the grassroots in a way that Weaver, who rankled the younger, digital-native Sanders campaign vets (a number of whom resigned when he was appointed last year), could not.

So does this mean Turner's potential run for Ohio governor is dead? Online groups have been agitating for her to run. A spokeswoman would only say (twice) that Turner "has not made a statement" on the matter.

Before you go: