Within the next eight days, the National Labor Relations Board plans to vote on a crucial rule that would speed up union elections—a change that’s drawn serious opposition from the US Chamber of Commerce and Republican politicians. Unions are very eager to see the law passed, since employers will often delay union elections until they can quash the organizing drive through intimidation or other means.

The NLRB has only three of five seats filled because of Republican obstruction in the Senate, and with two of those seats being held by Democrats, passage of the rule would seem likely. But there’s one way Republicans can stop this—and they might be on the brink of doing it.

In 2010, the Supreme Court ruled that the NLRB cannot function with only two members, as it did for a brief period at the end of President Bush’s term and the beginning of President Obama’s. The Court found that two members do not constitute a quorum, and invalidated everything the NLRB did during that time.

So in order to stop the union election rule, which is scheduled to be held by November 30, the lone Republican member can simply resign, and poof—no NLRB. This possibility was first flagged Monday by Mike Elk at In These Times, and today The Hill obtained a letter from NLRB Chairman Mark Pearce to the panel’s only Republican, Brian Hayes. The letter clearly indicates that Hayes has threatened to resign:

“In mid-October, I specifically discussed with you a potential schedule for consideration of the rulemaking. You did not offer any alternative schedule,” Pearce wrote in the letter to Hayes. “Instead, you indicated that, if the board proceeded with consideration of the matter, you would consider resigning your position.”

The obvious disadvantage of the resignation scheme, from a Republican point of view, would be that President Obama then gets an extra appointment to the NLRB. But Republicans have been so successful at blocking these nominations they might not be concerned about this possibility—and if Obama loses the 2012 election, it becomes a moot point.

In fact, Republican obstruction in the Senate might kill the NLRB at the end of this year anyhow. The term of Craig Becker, who President Obama put on the board with a recess appointment, will expire when Congress adjourns for the year. If Republicans can prevent Obama from making another recess appointment—as they’ve successfully done in the case of the Consumer Financial Protection Bureau—then the NLRB shuts down.

But for now, the goal is to kill the union election rule—and Republicans may be poised to hit the resignation kill switch.