This post was last updated at 6:28 p.m. EST.

Tierney Sneed contributed reporting.

A quartet of crucial Senate Republicans said they won’t back Senate leadership’s “skinny repeal” of Obamacare on Thursday unless they get a guarantee the House won’t just pass it into law, enough to kill the effort to repeal the law.

“There’s increasing concern on my part and others that what the house will do is take whatever we pass… go directly to the house floor, vote on it and that goes to the president’s desk,” Sen. Lindsey Graham (R-SC), flanked by Sens. John McCain (R-AZ), Ron Johnson (R-WI) and Bill Cassidy (R-LA).

“The skinny bill as policy is disaster. The skinny bill as replacement for Obamacare is a fraud,” Graham continued, referring to GOP leaders’ plan to pass a bill that eliminates the individual and employer mandates of Obamacare while touching almost no other parts of the law. “I need assurances from the speaker of the House and his team. … If I don’t get those assurances I’m a no.”

McCain concurred, saying “I am voting no” unless he has a guarantee of support.

Johnson made similar comments, saying “Just give us the assurance that whatever we pass tonight will go to conference,” while Cassidy was critical though less specific about his plan.

Afterwards, Johnson told reporters he’d been texting with House Speaker Paul Ryan (R-WI) and that Ryan knew they’d planned the press conference.

The senators hold plenty of leverage — if they vote no it dooms the bill, as GOP leadership has just two votes to spare heading into a late night vote-a-rama that will determine whether their partisan Obamacare repeal efforts keep going or fall apart.

But that guarantee isn’t something House leaders have yet been willing to promise — and it’s not exactly clear how they could make it ironclad since once the Senate passes a bill, the House can just send it along.

House Ways & Means Committee Chairman Kevin Brady (R-TX) told TPM shortly before the conference that the plan was to “gather with our members” and figure out next steps, repeatedly refusing to rule out passing whatever the Senate passes if they find it palatable enough.

“We’ll wait to see what’s coming out of the Senate and make a determination on conference or accepting it. So not yet known, really depends on what the Senate sends out,” he said.

Other senior Republicans were more explicit.

“They can demand anything they want but I’m not giving any commitment to any senator,” Rep. Tom Cole (R-OK) told reporters less than an hour before Graham’s comments, shortly after GOP leaders told members to keep their travel plans loose in case there are weekend votes.

“If we’re voting on Saturday, it would be the skinny bill,” said Rep. Chris Collins (R-NY), a close ally of President Trump who helped negotiate a part of the original House bill that won over some moderate New York Republicans.

And a spokeswoman for Ryan emailed just as the press conference was beginning that “Conference Committee is one option under consideration.”

Graham’s demands were a lot more specific than GOP leaders, who have put their good faith in House leaders to go to a conference committee where both sides would negotiate a plan both sides could pass.

“We’re going to get this bill out, and get it to them, and trust them to do the right thing,” Senate Majority Whip John Cornyn (R-TX) said Thursday afternoon.

But if conference fails and the Senate bill is the only option besides failure, it’s still easy to see how the House would decide to try to push through the Senate’s legislation.

Even Graham was unclear when asked what type of ironclad promise Ryan and company could make him to relieve his concerns enough.

“We don’t have it,” he said. “It’s like pornography: You know it when you see it.”