The House of Representatives is the epicenter of the dysfunction that grips the federal government. Much of it reflects the deepening polarization between the parties. But something else is at work as well—ill-conceived and obsolete rules that make polarization worse and prevent bipartisan coalitions from having their voices heard.

The speaker of the House is simultaneously too strong and too weak: too strong because his control of the Rules Committee allows him to dominate the legislative agenda and prevent individual members from influencing outcomes; too weak because he faces an ever-looming threat of removal from office by a small minority of his own caucus.

At present, the “Motion to Vacate the Chair” gives any individual member the right to call for the removal of the speaker and the election of a new one. It enjoys privileged status, meaning that it supersedes other pending business and goes to the head of the procedural queue. It is the Sword of Damocles hanging over the speaker’s head. The threat of such a motion was enough to induce John Boehner’s resignation. Paul Ryan requested but did not obtain the elimination of this motion as the price of accepting the speakership, a position he had not sought.

The Founders feared the tyranny of the majority. More recently, this power was concentrated in the majority of the majority. Since 2013, it has been wielded by a minority of the majority.

Last week, something remarkable happened: The Problem Solvers—a bipartisan House caucus of 24 Democrats and 24 Republicans—reached agreement on a package of reforms to break the gridlock, open up the legislative process, and give bipartisan coalitions a chance to have their ideas considered. Among the 10 proposals they endorsed:

The motion to vacate would be amended to require the support of one-third of the House to enjoy privileged status and 218 votes to pass.

Party ratios on almost all committees—including the Rules Committee—would reflect the party ratio of the entire House, preventing the majority from enjoying lopsided representation on key committees.

Closed rules, which prevent floor amendments, would require a three-fifths majority to pass. All other legislation would be taken up under open or structured rules.

Every structured rule would be required to allow at least one germane amendment from each party.

Under a structured rule, the Rules Committee must allow any germane amendment co-sponsored by at least 20 Republicans and 20 Democrats to reach the floor for consideration by the full House.

Bills with 290 co-sponsors or majorities of both parties would be marked up by the relevant committees and reported to the Rules Committee, which would report a rule unless a supermajority of the committee opposed it.

Although these proposals may appear to be obscure and technical changes, veteran House parliamentarians agree that they would transform the way the House conducts its business. Rank-and-file House members would have more say, and bills and amendments with bipartisan support would get to the floor for consideration by the full body.

After years of dispiriting dysfunction, it is easy to dismiss this rules package as one more well-intentioned but doomed gesture. In fact, the stars may be aligning for its success.

The most likely outcome of the midterm election is a narrow margin for one party or the other. Despite the Democrats’ lead in the national generic House ballot, the combination of geographical polarization and partisan gerrymandering leaves relatively few seats in play, and the distribution of Democratic votes leaves the party at a structural disadvantage. Despite a national House margin of eight percentage points in 2006, Democrats picked up only 31 seats, well short of the 63 seats the Republicans gained in 2010 with a smaller national margin. A 31-seat Democratic gain this year would leave the party with a 15-seat majority, about what the Republicans would enjoy under their best-case scenario.

In these circumstances, a small band of determined reformers can get its way by withholding its votes from the majority’s speaker-designate until he or she agrees to significant reform of the House rules.

It has happened before. In the election of 1922, Republicans lost 77 seats, leaving them with a slim majority. Led by Robert La Follette, frustrated Republican reformers seized their chance, refusing to vote for Frederick Gillett as speaker until the Republican leadership agreed to rules changes needed to open up the process. Party kingpin Nicholas Longworth scoffed, expecting the reformers to fold without a fight. Instead, they held their ground through nine ballots until the leadership relented.

Because the Problem Solvers are a bipartisan caucus, they are perfectly positioned to make their move, whichever party prevails in the House this year. The holdouts would come under enormous pressure from their party’s leadership, as the Republican reformers did almost a century ago. But if they stand their ground, they will prevail, quietly cheered on by other members who long have chafed under top-down leadership and restrictive rules that have denied them a voice in the legislative process.