“There is a huge onus on him now to show that this is the case and that he can bring people together to address the problems that he himself identifies as being an issue,” Zac Petkanas, a Democratic strategist, said of Mr. Manchin. Otherwise, “he is going to be personally holding up the things that he believes in.” If those efforts fall to Republican filibusters, Democrats hope Mr. Manchin and other reluctant party members will revisit their positions. “We do need to start testing this idea that the filibuster promotes bipartisanship,” said Senator Christopher S. Murphy, Democrat of Connecticut, who is trying to find 10 Republicans to back legislation imposing universal background checks on gun buyers. “I would hope that everyone is open to having their theories proved wrong if they don’t have evidence to prove their theories right.”... Regardless of Mr. Manchin’s position, Mr. Schumer said a vote would be called the week of June 24, as planned, “to protect voting rights and American democracy.”