Even the most optimistic interpretation of this week’s government shutdown quickly slides into the abyss of the bottomless distrust between Republicans and Democrats in the House.

The brief standoff over immigration and funding the government mostly dramatized Washington’s dysfunction. But it offered one glimmer of the hope when about two dozen senators from both parties coalesced to urge a quick resolution that reopened the government.

That loose alliance is now expressing optimism it can reach agreements on other thorny issues, starting with the long-term fate of the roughly 800,000 “dreamers” brought to the U.S. illegally as children by their parents. President Trump in September ended an Obama-era program that shielded them from deportation, though a federal District Court has temporarily blocked his decision.

Yet even if the Senate can come together on the “dreamers” or other issues, any such cooperation would confront the chasm between the parties in the House. GOP leaders there have already indicated they don’t feel bound to even consider the legislation if the Senate approves a deal.

Enter a new organization called With Honor.

The group, which is announcing its first campaign endorsements Thursday, has launched a major effort to elect to the House more recent military veterans who commit to working across party lines. The aim is to create a bipartisan core of House Members who are inclined to seek common ground, whatever their personal views. “The goal is to get enough of a critical mass that can function as a cross-partisan coalition and is able to get things done,” says Rye Barcott, the With Honor co-founder and CEO.