Over the long weekend, House impeachment managers huddled in House Speaker Nancy Pelosi’s office in the heart of the Capitol, hammering out the strategy to make their case against President Donald Trump on the Senate floor on Wednesday. They honed and filed documents arguing why their colleagues in Congress must vote to remove Trump from office, and took turns standing at the lectern on the Senate floor where they would make their historic case.

Handpicked by Pelosi to argue the House Democrats’ case in Trump’s impeachment trial, the group of seven impeachment managers—Reps. Jason Crow, Val Demings, Sylvia Garcia, Hakeem Jeffries, Zoe Lofgren, Jerry Nadler and Adam Schiff—include experienced litigators, prosecutors, and judges. Between them, they’ve been in public office for decades, taken part in at least three previous impeachment trials and logged time at the country’s top corporate law firms.

But in their jobs this week, they are in uncharted territory. As they begin their allotted 24 hours of opening arguments, they will try to convince lawmakers—and the public tuning in to the proceedings—that Trump’s actions are such a danger to American democracy that he must no longer serve as President. They are arguing that he not only abused the power of his office to withhold aid to a foreign ally in exchange for a political investigation into a rival, but that he upset the balance of power laid out by the founding fathers by refusing to cooperate in the House’s impeachment inquiry.

In an era of intense political polarization, that is no small task. To succeed, they must persuade at least 20 Republican Senators to take the difficult vote of convicting and removing a President from office who remains exceedingly popular with his base.

Politically, few in Washington are under the illusion that the seasoned group of lawyers and politicians can pull it off, regardless of how strong their legal arguments may be. Senate Majority Leader Mitch McConnell has been saying for months that he does not foresee a scenario where his majority splits to give the managers the votes they need need to convict Trump. This challenge was on display Tuesday, during 12 hours of debate about the rules that will govern the trial. Since Senators are prohibited from speaking on the floor when the trial is in session, it fell on the impeachment managers to advocate for the 11 amendments offered by Senate Minority Leader Chuck Schumer seeking to include witnesses and documents in the trial. Every vote failed along party lines.

“They know the subject matter, they have anticipated the opponents and they’ve been hitting all the right answers, but obviously [McConnell] has a grip on his caucus,” said Sen. Richard Blumenthal, a Democrat from Connecticut, during a break in the debate. “”We’ve known from the start they have an uphill battle.”

Their case is complicated further by the fact that a Senate trial is not a typical courtroom. “For most matters tried in civil proceedings you have pages of case law and thousands of instances where the law is applied,” said Philip Bobbit, a law professor at Columbia University. “We don’t have that in impeachment.”

Instead, the managers and Trump’s defense team will follow trial procedures passed on Tuesday by the Republican-led Senate. Those rules stipulate that lawmakers will only vote on whether or not to allow new evidence in the trial after opening arguments from both sides and a period during which Senators can submit written questions to Chief Justice John Roberts, who is presiding over the trial. Democrats have said they want to hear from at least four witnesses who evaded the House’s inquiry: former National Security Adviser John Bolton, acting White House Chief of Staff Mick Mulvaney, and White House officials Michael Duffey and Robert Blair. (Bolton has said he will testify in the Senate pursuant to a subpoena).

In order to allow more witnesses to testify, the 47 Democratic Senators —assuming they stay united—need to peel off at least four Republican Senators to join them in voting first to allow new evidence, and again to allow for witnesses.

“This is part and parcel of an effort to cover up the President’s misconduct,” Schiff, the lead impeachment manager, told reporters on Tuesday. “If McConnell makes this the first impeachment trial in history without witnesses or documents, it will not prove the President innocent. It will merely prove the Senate guilty of working with the President to obstruct the truth from coming out.”

Advocating for new witnesses carries its own risks. Under the new trial rules, if Democrats are successful in calling witnesses, Republicans will be able to issue subpoenas for people they want to hear from as well. Some Trump allies have said they want Hunter Biden, the son of former Vice President Joe Biden, to testify about his role on the board of a Ukrainian gas company, Burisma as a potential example of the corruption Trump asked about during a July 25 phone call with Ukrainian President Volodymyr Zelensky.

The managers are already pushing back on this possibility. “It would not be appropriate for the President to seek to call witnesses merely to try to perpetuate the same smear campaign that was foiled when his plot was discovered,” Schiff said Tuesday morning on CBS News. “Hunter Biden, for example can’t tell us anything about whether the President withheld military aid, whether he withheld that aid to coerce Ukraine to conduct political investigations, or why he wouldn’t meet with the President of Ukraine.”

Ultimately, the managers, operating in a Republican controlled-Senate, have little control over the witnesses that get called. But they’ve made clear that they will use the leverage they have to make their case.

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Write to Alana Abramson at Alana.Abramson@time.com.