It’s the political equivalent of “Who’s on First?” that has left GOP leaders in Congress squabbling with one another. And it’s an internal political battle that could come to a head this week.

In the wake of Speaker John Boehner’s (R-Ohio) announcement that he plans to retire from Congress at the end of October, rather than finishing out his 13th term, the GOP has found itself without a consensus choice to lead the party in the U.S. House of Representatives. GOP leadership had initially coalesced around House Majority Leader Kevin McCarthy (R-Calif.), but pressure from members of the Freedom Caucus — a right-wing group of 30 to 40 Republicans who want the GOP to be tougher on spending cuts and opposing President Obama — forced McCarthy to announce two weeks ago that he was dropping out of the race.

Such is the problem with any candidate for Speaker. So long as members of the Freedom Caucus insist on one of their own — or at least a leader who will prioritize pushing conservative legislation and play hardball with the White House — they have enough votes to not only deny a potential Speaker the votes for confirmation, but threaten rebellion and repercussions for any top Republican who cuts a political deal with the Democratic White House.

According to most news outlets and Beltway conventional wisdom, U.S. Rep. Paul Ryan (R-Wisc.), the party’s vice presidential nominee from 2012, has been touted as a figure that could be acceptable to both the ultra-conservative flank and the more establishment Republicans within the GOP caucus. However, Ryan is reportedly skeptical of taking the job in light of Boehner’s resignation and the Freedom Caucus’ threats.

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Additionally, some right-wing organizations and media outlets have attacked Ryan as insufficiently conservative due to his votes in favor of various bailouts during the financial crisis of 2008 and the subsequent recession, for cutting a budget deal with Senate Democrats, and for his support of comprehensive immigration reform.

According to CNN, Republicans were expected to hash out their differences this Wednesday, when the GOP was scheduled to hold two closed-door conferences to discuss the party’s future and its vision for how the House will be run. Some proposals that are expected to be discussed include: placing more Tea Party Republicans in positions of leadership, stopping retaliations against those conservative members who buck party leadership on key votes, and pushing for votes on conservative legislation, even those bills which either do not have enough support to pass or will be vetoed by President Obama.

In light of the most recent political developments, accurately predicting the next Speaker of the House is essentially as reliable as a political parlor game. Besides Ryan, other names that have been floated include Reps. Bill Flores (R-Texas), the head of the Republican Study Committee, another conservative caucus within the GOP, Lynn Westmoreland (R-Ga.), Daniel Webster (R-Fla.), Peter Roskam (R-Ill.) and Marsha Blackburn (R-Tenn.). In all, the names of more than 20 different members of the GOP caucus have been floated as potential speakers, but some of those mentioned are considered too close to the current party leadership, likely dooming their chances of emerging as a consensus candidate.

As a result of the unpredictability surrounding the Speaker’s race, it is also unclear whether the next GOP House leader would be as willing to consider fostering a working relationship with LGBT conservatives as Boehner has, let alone allowing votes on pieces of pro-LGBT legislation.

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“I, at this juncture, am not going to make any statements expressing favoritism for any particular candidate for Speaker,” says Gregory T. Angelo, president of the Log Cabin Republicans, when asked for political predictions. “This is still a very fluid race, and I would want to make sure that whomever gets the speakership starts off on the right foot with Log Cabin Republicans, and that we’re able to have a working relationship with the Speaker’s office, in the same way we’ve had a working relationship with Speaker Boehner’s office in the past.”

But David Stacy, the director of government affairs for the Human Rights Campaign, says that those looking for a positive outcome from the Speaker’s race, at least in terms of LGBT rights, are likely to be disappointed.

“None of the candidates for Speaker have particularly good records on LGBT issues,” Stacy says. “I do think you can differentiate Paul Ryan a tad, because he voted for a sexual orientation-only version of ENDA in 2007, and recently voted on an amendment on the Transportation and Housing appropriations bill that said no funds can be used in contravention of the president’s federal contractor executive order.”

That said, Stacy adds, the rest of Ryan’s record on LGBT rights — especially on issues related to marriage and relationship recognition — is not especially favorable. According to HRC’s congressional scorecard, of the 20 to 25 members whose names have been floated as a potential Speaker, almost all of them, including Ryan, have zero ratings.

But Angelo dismisses doomsaying from the left, saying his LGBT organization is hoping to establish working relationships with whomever becomes the next Speaker in order to lobby the GOP on embracing pro-equality measures.

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“I would just say this: there has rarely been a door that has been closed to Log Cabin Republicans,” Angelo says. “And I can only speak for my time as president of this organization, but there has rarely been a time that a Republican office has not expressed a willingness to engage with us, discover and learn more about the issues for which we’re lobbying, and find out more about what it is to be an LGBT Republican.”

Angelo acknowledges that the next Speaker will have a lot of work on his or her hands to unite the party and reconcile divisions among the various factions within the GOP caucus. He also has praise for Boehner, whom he said did “yeoman’s work” and was never credited for holding together “one of the most fractured House Republican conferences in history.” He believes the person who will emerge as Speaker will be the one who articulates a clear vision of what they hope to accomplish while promoting conservative ideas and values.

“I think the thing that’s going to ultimately help the party and the House to forge ahead is someone who is able to take the Speaker’s gavel and come up with a new and innovative way to lead this caucus,” he says. “Ultimately, some sort of change is needed in terms of the way leadership approaches legislation, approaches passage of legislation, and so I think there is some opportunity there for a consensus candidate to emerge. But they can’t just be a consensus candidate; they need to be a consensus candidate that presents a vision for how the day-to-day work of the House is going to transpire.”