In his opening remarks on Tuesday morning, Senate Majority Leader Mitch McConnell listed various ways the body should “provide a lasting tribute” to John McCain.

One idea would be to rename the Armed Services Committee room after him. The late Ted Kennedy, McConnell explained, got a room too. Or they could hang his portrait in the Senate Reception Room. Only a few senators, including McConnell’s idol Henry Clay, have received such an honor, he noted. Portrait hangings are meaningful! McConnell further announced that he would put together an official, bipartisan group of senators to determine the appropriate commemoration.

What McConnell did not mention in the list of ideas, however, was the resolution introduced by Sens. Chuck Schumer and Jeff Flake to rename the Russell Senate Office Building, currently named after the late, powerful Georgia Sen. Richard Russell, after McCain. He had reason not to.

Other Republicans were surprised to find themselves in agreement with the Senate minority leader.

It’s become apparent enough over the past day that this might not go over so well with segments of the Senate Republican conference, particularly those from the South. For some senators, renaming the Senate Office Building would not just be about honoring McCain but about stripping an honor from Russell, a famed segregationist. This sort of renunciation of troublesome history is anathema in Trump’s GOP, where battles to save Confederate monuments from progressive demands that they come down have become almost commonplace. Were the Schumer-Flake resolution to linger in the ether as the only available option out there, with pressure to act upon it growing, it might blossom into an ugly fight with racial dimensions that McConnell doesn’t want to deal with. So McConnell did what government leaders do when what seems like an uncontroversial measure at first displays symptoms of future ugliness: He kicked it to a group for further study.

It’s not that Republicans are uniformly against renaming the building. Flake and other departing senators are on board. “I’m fine with it,” Tennessee Sen. Bob Corker told me Tuesday. Utah Sen. Orrin Hatch’s office said Monday that “it’s a worthy goal” and he “looks forward to working with the majority leader to accomplish it.”

Other Republicans—including some who were staying in the body—were surprised to find themselves in agreement with the Senate minority leader. “I would say this is an idea of Chuck Schumer’s I actually like,” Wisconsin Sen. Ron Johnson told me. “I think it’s certainly deserved, and I’m certainly not opposed to it.”

As for the rest? They won’t say that naming the building after John McCain is a bad idea, just that naming a room or hanging a painting might be better. Somehow.

“I think there’s a number of ideas that have been suggested,” South Dakota Sen. Mike Rounds said. “I do like the idea of following almost like a regular order, where we get a bipartisan committee to really kind of look through all the different ideas that are out there, and then come up with a plan that would have good solid bipartisan support.”

I asked him what the argument was against renaming Russell for McCain.

“Oh I don’t think there’s an argument against it,” he said. “I think it’s just a matter of, rather than just taking the first [idea] that pops, let’s put that into the mix with everything else, without prejudice, and look at all the different ideas.”

But, again, there is an argument against it. It has nothing to do with giving McCain a building. It has to do with taking one away from Russell.

Richard Russell was a leader of segregationist Democrats who obstructed civil rights legislation for years. He was, as even his 76-year-old niece, Sally Russell, told the Washington Post, a racist. But he was also, she went on, “much more than that,” citing his authorship of the National School Lunch Act, support for New Deal programs, and his leading voice in foreign affairs as longtime chairman of the Armed Services Committee.

These were some of the same arguments that current Georgia Sen. David Perdue made to reporters on Tuesday in explaining why he believed that Russell “absolutely” deserves to still have his name on the building. (Even though, he made clear, that this position “doesn’t say anything about” his feelings on John McCain.) It was interesting listening to a conservative Republican try to run through the domestic achievements of a Democratic “giant” without implying that he liked any of those actual positions.

“He was a big supporter of the Great Society, the War on Poverty,” Perdue said. “We all know that those things failed, but he was a big champion of them.” Russell wasn’t actually a big champion of those programs. Would that make Perdue like him … less? More?

One thing was clear, at least.

“This renaming thing, because of one issue,”—decades of leading legislative resistance to civil rights—“is somewhat troubling,” he said.

This is how the issue could get ugly: If a very real debate about renaming a monument—in this case, the building in which many of them work—to a racist reached the Senate floor as the Senate is unifying in grief, some Republicans would feel compelled to defend said racist.

“I think we ought to be careful what we do,” Alabama Sen. Richard Shelby told reporters. “There are a lot of ways to honor people. I never thought it was a good idea to name [something for] somebody who was a historical figure … and then change it to something else because times change.”

This would mean the debates over “erasing history” and seeking to undo honors bestowed upon avowed white supremacists spilling into highly public view on the Senate floor over what should be the unifying issue of honoring a Senate legend’s passing. For now, it will be had in the comfort and privacy of a task force instead.