Key conservatives in the House are warning potential candidates for speaker not to expect their support unless they commit to giving them more power in the legislative process.
Members of the House Freedom Caucus -- many of whom were elected to Congress during the 2010 Tea Party wave -- want to decentralize power in the GOP leadership circle so that conservatives have a better shot at influencing policy and process.
They only make up roughly 40 Republicans out of the lower chamber's 247, but they have the power to throw up more roadblocks if they don't coalesce around the next leading candidate for speaker.
While they haven't issued an official ultimatum as a group, a number of conservatives say they want the next speaker to protect members from being punished if they disobey leadership and allow a more open process for bringing bills and amendments to the floor.
Assuming no Democrats vote for the leading GOP candidate for speaker, and the Freedom Caucus also votes en bloc against that lawmaker, that candidate wouldn't win the 218 votes necessary to become speaker.
And that what House Majority Leader Kevin McCarthy, R-California, was concerned about last week before he dropped his bid.
Last week, Politico obtained a six-page questionnaire prepared by the Freedom Caucus to spark a conversation among members about how candidates would handle different situations as speaker.
"Would you ensure that the House-passed appropriations bills do not contain funding for Planned Parenthood, unconstitutional amnesty, the Iran deal and ObamaCare?" one question said.
Over the last year, conservatives have lobbied GOP leaders to use their majority advantage to torpedo the Iran deal, repeal ObamaCare and defund both Planned Parenthood and President Obama's immigration executive actions. Those last two efforts, however, backfired and catapulted the GOP into multiple shutdown fights with Democrats.
On many issues, conservatives accused Speaker John Boehner, R-Ohio, of eventually capitulating to Democrats and President Obama and, as a result, they repeatedly threatened to oust him.
Now that Boehner is resigning, they are pleading for and expecting change.
And members of the Freedom Caucus are not alone. The conservative Republican Study Committee is wading into key policy fights. In September, it unveiled a government spending package that would directly target President Obama's policies.
Later this month, just before the critical Nov. 5 debt limit deadline, the Republican Study Committee plans to release legislation that would raise the ceiling, but it will also call for mandatory funding cuts in exchange for averting default. Many conservatives say they oppose a "clean" debt ceiling increase (that is, without conditions), which they fear Boehner will introduce before he leaves.
They want regular order
On CBS's "Face the Nation," last Sunday, Rep. Mick Mulvaney, R-South Carolina, said his Freedom Caucus peers are more concerned about the next speaker's commitments than his record.
"So we're looking for more of a process and principles than we are a person," Mulvaney said.
Even more than policy proposals, conservatives are demanding "regular order" in the House whereby lawmakers have the freedom to offer amendments to bills, have more power in committees and can pass all 12 spending bills on time each year.
"Paul certainly has a lot of respect across the Republicans in the House. I think he could also be a good Speaker," Mulvaney said.
Rep. Paul Ryan, chairman of the tax-writing House Ways and Means Committee, has so far declined to run. But now, with no other consensus leader in sight, he's under extreme pressure from Republicans - both leadership and rank and file. His spokesman repeatedly said this week that he had no updates on Ryan's decision-making process and said not to expect to find out this week, while Congress is out of session.
So far, the Freedom Caucus has only endorsed Rep. Daniel Webster, R-Florida, for speaker. Before arriving on Capitol Hill in 2013, he served in the Florida legislature for 28 years and served for two of those years as House speaker.
"We're looking for a bottom-up approach that involves more members, that is consistent with conservative principles," Rep. Tim Huelskamp, R-Kansas, said in an phone interview Wednesday with CBS News.
Huelskamp, chairman of the Tea Party Caucus, said the next speaker "can't be John Boehner 2.0."
"This is not a coronation, this is not a crowning, this is not 2010 again when they all move up the ladder," he said.
Since January, Boehner has retaliated against conservatives who have taken action against his leadership. He kicked two Republicans off the powerful House Rules Committee after they voted against him for speaker. Another Republican was later stripped of his subcommittee gavel after he voted against a procedural motion on trade bills.
Rep. Mo Brooks, R-Alabama, in a phone interview with CBS News on Tuesday, said he wants a new rule guaranteeing members won't be stripped of their chairmanships or committee assignments if they actively defy leadership.
Brooks said he would be open to Ryan as speaker despite what he called Ryan's "troubling" record on immigration reform and budget issues.
"Paul's record has clearly established that he is in the liberal half of the Republican conference. So is Daniel Webster. So is McCarthy," Brooks said, adding, "I could be supportive of a Republican who's from the liberal half of our conference, provided that there are other things that the liberal Republican will do to make my district feel more comfortable."
House Republicans will continue discussing the leadership race when they return to Capitol Hill next week. Last week, Boehner said he was confident it could be resolved within a matter of weeks.
It's also possible Boehner could remain as speaker until January 2017, said Brooks, who added that he wouldn't have voted to oust him.
"The risk is if John Boehner leaves...we will get a speaker who is even more liberal and less in tune with districts like mine," he said.