(CNN) As a dozen Democratic presidential candidates descend on Ohio for their debate on Tuesday, a larger political question looms: Is Ohio still a winnable swing state worth fighting for in the general election?

Yet the prospect of abandoning Ohio, which Barack Obama won twice, is deeply offensive and wrong-headed to party leaders here.

"It's absolutely a mistake to write it off," Ohio Democratic Sen. Sherrod Brown told CNN. "It's always gone with the winner and it will again next year."

The last Democrat to win the presidency without capturing Ohio is John F. Kennedy. But in the intervening 60 years, every Democrat who made it to the White House has carried Ohio, a trend that Brown believes will almost certainly continue.

Interviews with more than two dozen party leaders, strategists and Democratic voters here suggest the true answer to whether Ohio remains a competitive battleground revolves largely around who wins the primary battle.

It's less a question of the candidate moving in a progressive or moderate direction, party leaders say, but rather of policies that may be portrayed as unrealistic or extreme.

"I think it's a terrible mistake if the Democratic nominee would publicly support 'Medicare for All,' " said Brown, an unabashed progressive who believes such a policy could scare voters and unwittingly give Trump a political advantage on health care. He urged candidates to focus on improving the Affordable Care Act, while protecting pre-existing conditions.

David Pepper, chairman of the Ohio Democratic Party, takes pains to not take sides in the 2020 primary. But he joins Brown in arguing that Medicare for All -- and ultimately abolishing private health insurance plans -- could turn off many voters here.

"People don't want to be told they're going to lose their health care or their health care plan," Pepper told CNN. "Whoever the candidate is, there are ways to broaden health care without having people feel like their own health care situation will be thrown upside down."

That tension will be front-and-center Tuesday night at Otterbein University as 12 Democratic candidates take the debate stage. Sens. Bernie Sanders and Elizabeth Warren are among those who have championed Medicare for All, while former Vice President Joe Biden, South Bend, Indiana, Mayor Pete Buttigieg and others have questioned the disruption such a sweeping policy would pose.

It's one of the biggest differences among the Democratic rivals. The outcome of the policy fight could play a large role in determining who becomes the party's nominee.

In 2020, winning Ohio will be no small task for Democrats. While Obama carried the state in 2008 and 2012, Trump's 8-point victory over Hillary Clinton showed how deeply red Ohio can be.

But the suburbs are changing. Here in Westerville, just outside Columbus, Republican support has fallen off. In 2012, Mitt Romney beat Obama 53% to 45%. Four years later, Clinton won 50% to 45%, despite losing the state overall.

"This new block of voters in the suburbs we have to work hard to keep. We shouldn't take for granted that they're just going to vote for us," Pepper said. "But Ohio can be the state that ends the Trump presidency."

Jane Timken, chairman of the Ohio Republican Party, said she believes Trump is in a strong position as he stands for reelection. The President has visited the state multiple times, is building a large campaign operation here and is investing heavily in branding Democratic candidates as too radical for Ohio.

"Those messages don't really resonate in Ohio, and I don't think suburban voters are going to buy it," Timken told CNN.

Yet that remains an open question, which is impossible to answer with any certainty one year before Election Day. One of the biggest things that has changed from the last campaign: the Trump presidency is now motivating Democrats in a way that it did not four years ago.

Nan Whaley, the Democratic mayor of Dayton, said voters should hold Trump accountable for his trade policy, promises to restore manufacturing jobs and for failing to swiftly take action on guns, a key issue after an August mass shooting killed nine people and injured 27 in her city.

"Why would you abdicate Ohio so quickly to Donald Trump?" Whaley told CNN.

"We really need to pay attention to the messaging around suburban voters -- particularly women," she added. "Hillary's loss awakened a group of people that were not interested or willing to do the work of politics who are now completely fixated because they know what's at stake in 2020."

Whaley is talking about women like Stephanie Pyser, Tiffanie Roberts and Lisa Ludwig, who were awakened by Trump's victory and have become increasingly angered by his presidency. A few weeks after the 2016 election, they decided to form a group they call 'Positively Blue.'

They've been organizing like-minded voters ever since from their kitchens and front porches in Dublin, Ohio, a suburban community just outside Columbus.

"This neighborhood tends to be a little bit more Republican, so I wouldn't have brought up politics too much at social settings," Pyser said. "That was one of the reasons we started Positively Blue because we didn't have anyone to talk to."

They're motivated by a sense of obligation -- and perhaps lingering feelings of guilt -- for not being more active in the last presidential race.

"I wish I would have been more involved prior to the 2016 election. Truthfully, I didn't think that that would be the outcome," Ludwig said. "I like to think I'm pretty in tune, but I never saw it coming."

Now, their group has 310 members. They've been working to elect local candidates, but soon will turn their attention to the presidential race.

"Going into 2020, we're full steam ahead," Pyser said. "That's when our members really step up and want to get more involved in canvassing, fundraising and all the many things that we get to do. We'll be ready."

Katie Paris said she hears similar sentiments across Ohio. From her home in the Cleveland suburbs, she started a group called Red Wine and Blue to bring together women from all corners of the state and help them become politically active.

"Little did I know I would find this everywhere across Ohio," Paris said. "Women are taking action and they're talking to their friends and that's how change happens."

Paris, a Democratic consultant who spent years in Washington, moved to her husband's native Ohio in 2012 and is helping organize suburban women in advance of the 2020 election.

She dismisses questions of whether Ohio is still a pivotal swing state, pointing to a handful of key suburban legislative races Democrats won in 2018. She said groups like Priorities USA, one of the largest Democratic super PACs that did not include Ohio on its list of top 2020 priorities, are giving up too easily because of the 2016 defeat.

"It's absolutely a mistake to overlook Ohio," Paris said, standing on a front porch on a recent fall afternoon. She said her message to the Democratic nominee was simple.

"Spend time here," Paris said. "Trump sure is. What does that tell you? If he was sure he had Ohio locked up, would he be coming here so often?"

Hours before the debate on Tuesday -- and just miles away -- the President's reelection campaign is holding a "Women for Trump" event in Columbus. It's one sign, at least, that Ohio -- and suburban women voters -- are among the highest targets of the 2020 campaign.

CLARIFICATION: This story has been updated to clarify Katie Paris' move to Ohio.