A bipartisan advocacy group is launching an ambitious effort to overhaul how the House conducts legislative business, with the goal of breaking through the partisan gridlock that has long paralyzed Congress.

The organization, No Labels, will kick off discussions about the so-called Speaker’s Project on Thursday during a luncheon at the Library of Congress featuring several members of the bipartisan Problem Solvers Caucus. Former House Majority Leader Richard Gephardt (D-Mo.) will also be in attendance.

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The idea behind the project is to encourage lawmakers to withhold their support for the next Speaker unless the candidate agrees to a package of rule reforms designed to foster more bipartisanship and participation from rank-and-file House members.

No matter which party wins control of the House in the fall, they are expected to have a slim majority, meaning that a small group of lawmakers could have major sway over the selection of the next Speaker if they band together to demand changes.

That scenario leaves bipartisan advocates like No Labels believing that Congress faces its best chance yet to fix what many frustrated lawmakers have described as a broken political system in Washington.

“What got us really excited is that the environment is shaping up in a way where after this election, you’re going to have pretty narrow control, one way or another,” said Ryan Clancy, the chief strategist at No Labels. “There is actually an opportunity to exert leverage.”

The organization is planning an aggressive grass-roots campaign to educate the public about the Speaker’s Project. They hope that will lead constituents to urge their members to get on board with the effort, which will need buy-in from lawmakers.

The idea has already caught the eye of some members, including Rep. Tom Reed Thomas (Tom) W. ReedCentrist House group offers bipartisan COVID-19 relief deal House approves two child care bills aimed at pandemic Diabetes Caucus co-chairs say telehealth expansion to continue beyond pandemic MORE (R-N.Y.), the co-chairman of the Problem Solvers Caucus. Reed has previously said he would consider supporting a Democrat’s bid for Speaker if they agreed to certain rule reforms.

Some potential reforms that No Labels is pushing include allowing more open rules and amendments, ensuring conference committees have members from both parties, eliminating a tool that allows any member to force a vote to oust a sitting Speaker and requiring that a lawmaker receive a larger majority to win the Speaker’s gavel.

No Labels argues those changes would force the majority to negotiate with the minority and increase the odds of achieving bipartisan solutions in Congress.

Any changes to loosen the Speaker’s grip on power are sure to face fierce resistance in both parties, and it’s unclear whether enough centrist lawmakers would be willing to play hardball.

Still, advocates note there is some precedent for success. In 1923, a group of progressive Republicans demanded congressional reforms and withheld their votes for Speaker until they were successful.