It’s the most popular parlor game on Capitol Hill: Just how many Republican losses will it take before House Minority Leader John Boehner (R-Ohio) and other leaders get axed? Ten? Twenty? Somewhere in between?

Generally speaking, House Republicans told Roll Call that there will be grumbling if the number is in the double digits. If GOPers keep the number under 10, the thinking goes, the current leadership team has nothing to worry about. Anything from 10 to 20 lost seats could spark a challenge, while defeats piling up past 20 would nearly guarantee a clean sweep.

Complicating the picture is that there are no obvious challengers. No one, even Chief Deputy Minority Whip Eric Cantor (R-Va.), has an easy path to topple either Boehner or Minority Whip Roy Blunt (R-Mo.).

“You can’t beat somebody with nobody,” said one conservative lawmaker who advocated a wholesale change after the 2006 debacle and still wants new leadership.

Boehner, meanwhile, has helped consolidate his hold and given the party hope for the fall with his narrow focus on oil drilling. “The surge on the energy issue has been a huge feather in John Boehner’s cap, and people are starting to feel really good about things,” another Republican lawmaker said. “It was really manufactured — it didn’t just happen — and he gets credit for it.”

Most Republicans also credit Boehner for aggressively attacking Ways and Means Chairman Charlie Rangel (D-N.Y.) for his ethical problems and trying to make the embattled lawmaker a poster boy for a Democratic Party that GOPers charge has failed to deliver on its promises to reform Congress.

His offensive against Rangel — a friend, by both men’s accounts — has helped allay fears in the Conference that Boehner lacks the backbone to get tough with Democrats. On a near-daily basis, Boehner’s office has called for Rangel’s ouster as chairman of the tax-writing panel. After more than two dozen Republicans broke ranks last month to support Rangel on a measure Boehner offered to rebuke him, the GOP leader threatened the committee assignments of any who defected again. Last week, Boehner put the matter to a vote again, and only five Republicans strayed.

“If leadership had failed to do something, if leadership was perceived as a lap dog for Democratic leadership and not offering enough opposition, that would be one thing. But that’s not the case,” Rep. Jack Kingston (R-Ga.) said.

Republicans are pointing to another factor giving them hope heading into November: Alaska Gov. Sarah Palin, their vice presidential candidate. She has energized what was expected to be a dispirited conservative base, raising the hope of a downballot effect that could save some of their most vulnerable contenders. “All down the line, it’s given a boost to our Congressional candidates,” Rep. Phil Gingrey (R-Ga.) said.

Those factors together have lifted House Republicans from their spring doldrums, when they suffered a devastating string of three special election losses in what should have been friendly territory and faced both a campaign committee in disarray and a listless presidential candidate.

But GOP leaders are aware that rising expectations could also backfire. If Republicans get their hopes up only to witness an underwhelming performance on Election Day, they will be more inclined to agitate for new blood in their leadership ranks. “If a stockbroker loses you 5 percent, and the market loses 20 percent, you’re not as mad at your stockbroker,” Kingston said.

To that end, leaders are reminding the Conference that Republicans still face a grim political environment. The Democratic Congressional Campaign Committee is sitting on $54 million in cash, four times that of its Republican counterpart. While Democrats are defending only three open seats, Republicans are fighting to hang on to more than two dozen. And if the turmoil in the financial markets proves a major voting issue, it will likely play to the advantage of Democrats, who take a tougher line on Wall Street and deregulation.

“We’ve got to be realistic about our goals, and I think everyone understands that,” one GOP leadership aide said.

For Republican leaders watching their backs, a key question remains: Who is in a position to launch a bid? Cantor, though a member of leadership himself, ranks at or near the top of any list. Earlier this month, he unveiled to Roll Call a middle-class agenda he said had been in the works for a year. Though Cantor presented his plan as a supplement to a Boehner-authored blueprint from the spring, the rollout caught other leadership offices off-guard — and raised eyebrows.

In an interview that day, Cantor acknowledged the elections could shake up the leadership team, though he declined to name a threshold number of losses. “What will happen post-election, I believe that it really is about what narrative we tell over the next five weeks,” he said.

Spokesmen for Republican leaders are likewise declining to speculate on what election results might mean for their bosses’ careers.

“We know we’ve got an uphill fight, but we’re going to the country with a message on energy and the economy that the American people support,” Boehner spokesman Michael Steel said. “The whole leadership team is entirely focused on helping Republican candidates win on Nov. 4. We’re not thinking about anything after that.”

Blunt spokeswoman Antonia Ferrier said the Missouri Republican “has been honored and privileged to serve as whip. He takes his job very seriously and works hard to earn the respect and trust of his colleagues each and every day. Mr. Blunt is rightly focused on serving the House Republican Conference and not on rumor and speculation.”

Cantor notwithstanding, several lawmakers and aides said the incumbent team can draw comfort from one fact: Their most serious competition, a Republican with the energy and intellectual heft of former Speaker Newt Gingrich (R-Ga.), has yet to emerge.

“There’s not anybody like that,” Rep. Mike Simpson (R-Idaho) said. Echoed a GOP aide, “We do not have a Newt Gingrich.”

But Rep. Ray LaHood (R-Ill.) suggested that while the next generation of Republican leaders might not be known today, it could emerge from the wreckage of a devastating November.

“There’s a lot of ambitious people in leadership positions now below Boehner,” he said. “If one person gets challenged, every leadership post will be up for grabs. It’s sort of like a house of cards. Once Members sense one is being challenged, there will be blood in the water.”