Washington (CNN) In the six weeks since Justice Ruth Bader Ginsburg revealed she had endured another cancer ordeal, she has made more than 10 public appearances, animated and "very much alive." Yet advocates with a stake in another Supreme Court confirmation battle cannot help but be on edge.

If the 86-year-old, four-time cancer survivor were to leave the bench at any point during the rest of a Trump presidency, it would set in motion a succession fight like no other. President Donald Trump would be securing more than a third lifetime appointment to the high court. He would be replacing an unerring liberal, a women's rights icon, with a conservative.

That would thrust the court rightward in a way not seen since 1991 when conservative Justice Clarence Thomas succeeded Thurgood Marshall, himself a civil rights legend. The court already has a 5-4 conservative advantage.

Such stakes for the law have necessarily roused many in the nation's capital as well as people across the country following the moves of the "Notorious RBG" and offering public wishes for her good health.

Ilya Shapiro, director of the Cato Institute's constitutional studies, recently recommended the White House update its list of possible Supreme Court replacements, citing Ginsburg's latest cancer scare. The list of 25 potential Supreme Court candidates was last updated in November 2017 and was one way Trump back in 2016 sought to assure Republican voters he would appoint a strong conservative to the bench.

"It would be both smart politics and good governance to update the list," Shapiro wrote on The Federalist, a conservative website.

Liberals, meanwhile, are preparing their own list of recommendations for the high court, hoping that the next vacancy does not occur until a new president is elected in November 2020, and that the winner is a Democrat.

"The fight to replace Justice Ginsburg would have a different tenor to it," said Christopher Kang, chief counsel of the liberal advocacy group Demand Justice, yet he adds, "For us, the thing that engages progressives far more than RBG's prominence is Brett Kavanaugh's sheer presence on the court."

As Demand Justice compiles a list of possible nominees to recommend to Democratic presidential candidates, it is organizing with other liberal groups a protest at the Supreme Court for Sunday to mark the one-year anniversary of Kavanaugh's Senate confirmation.

Trump continues to vigorously defend Kavanaugh, whom he appointed to succeed centrist-conservative Justice Anthony Kennedy last year. Trump used Twitter to swat back recent criticism triggered by renewed complaints of sexual misconduct by Kavanaugh in his youth. Kavanaugh has categorically denied the accusations.

In a September 15 tweet, the President declared of critics, "They are trying to influence his opinions. Can't let that happen!" The next day Trump added, "Do you believe what these horrible people will do or say. They are looking to destroy, and influence his opinions -- but played the game badly."

If a high court seat were to open over the next year, Senate Majority Leader Mitch McConnell has said he would not adhere to his 2016 election-year stance. After Justice Antonin Scalia died suddenly on February 13, 2016, and President Barack Obama nominated US Appeals Court Judge Merrick Garland, McConnell blocked any consideration of Garland, saying the next president should name Scalia's successor. (Trump chose Neil Gorsuch in 2017.)

McConnell in September told radio host Hugh Hewitt he would "absolutely" move on a nominee in the 2020 election year. McConnell has differentiated his 2016 stall from possible 2020 action by noting that the White House and Senate are now controlled by the same political party. This time, McConnell has observed, there would be no expected conflict between his Republican majority and a Democratic White House.

Senate Minority Leader Chuck Schumer and other Democrats labeled McConnell a "hypocrite" earlier this year when he made his new position clear.

"There was nothing I did (in 2016) that ... would not have been done had the shoe been on the other foot, had there been ... a Republican president and a Democratic Senate," McConnell insisted to Hewitt. "So look, they can whine about this all day long. But under the Constitution ... the President makes the nomination, and the Senate confirms."

Last term for the current nine?

As the 2019-2020 Supreme Court session begins, an imminent question is whether this could be the last for these nine justices.

Ginsburg is one of three justices who are over 70. Stephen Breyer, who is also an appointee of Democratic President Bill Clinton, is 81, and Thomas, an appointee of Republican President George H.W. Bush, is 71.

Neither Ginsburg nor Breyer have displayed any interest in retiring. Thomas might be most inclined to leave the bench while a Republican holds the White House. But Thomas, now the leader of the far right of the court, has told close associates he hopes to remain for many more years.

Ginsburg said that her travel and speaking agenda enhances her healing. "It's a necessity to get up and go," she said in a September appearance in New York City . "It's stimulating, and somehow all these appearances that I've had since the end of August, whatever my temporary disability is, it stops and I'm OK for the time of the event."

Since the Supreme Court revealed on August 23 that a malignant tumor on her pancreas had been detected and treated, Ginsburg has spoken in Buffalo, Washington, DC, New York City, Little Rock, Raleigh and Chicago, some places multiple times.

List-keeping

A White House spokesman said this week that the administration would have no comment on whether Trump's list of Supreme Court candidates was being updated. He also declined to comment on any other matters related to the administration's readiness for any Supreme Court opening.

Leonard Leo, a Federalist Society leader who has steered multiple GOP Supreme Court selections, including that of Kavanaugh, told CNN recently, "I think it's inappropriate to be having 'what if' discussions about Justice Ginsburg's health. But the fact that the left is talking so much about Justice Ginsburg demonstrates her importance to them as a symbol and the intensity of the kinds of campaigns they would wage over Supreme Court vacancies in the future."

Among the people on the Trump Supreme Court list who met with the President during past selection processes and could be likely finalists are US Appeals Court Judge Amy Coney Barret of the Chicago-based 7th US Circuit Court of Appeals and 6th US Circuit Court of Appeals Judge Amul Thapar, a Kentucky favorite of McConnell.

Demand Justice's Kang said the group is compiling a list to be released by the end of the month. Kang and Brian Fallon, who with Kang co-founded Demand Justice in 2018, wrote in an The Atlantic that Democratic presidents should look to people who have represented workers, consumers or civil-rights plaintiffs, rather than corporations. (Kang and Fallon both worked in the Obama administration.)

Among the names likely to be offered, based on Kang and Fallon's prior statements, are Sharon Block, a former National Labor Relations Board member, Pamela Karlan, a Stanford law professor, US Appeals Court Judge Cornelia Pillard of the DC Circuit, Bryan Stevenson, director of the Alabama-based Equal Justice Initiative, and Jenny Yang, a former chair of the US Equal Employment Opportunity Commission.

"The list is not, literally, about one person and one vacancy," Kang said. "It's about how to approach all vacancies" on the Supreme Court and throughout the three-tiered federal judiciary.

McConnell has made a high priority of confirming Trump nominees to all levels of the federal bench, and he told Hewitt last month: "We're not going to leave a single vacancy behind by the end of next year."