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They found he placed toward the back of the pack among the 2016-ers — behind the likes of Mr. Bush, New Jersey Gov. Chris Christie and Rep. Paul Ryan of Wisconsin, the party’s 2012 vice presidential candidate.

For his part, Mr. Ryan has kept his cards close to the vest in the Obamacare fight, telling The Wichita (Kan.) Eagle newspaper this week that he is waiting on the Senate to act before he makes up his mind.

“If they don’t repeal Obamacare, defund or delay it, then we’ll adjust our strategy accordingly and keep pressing for it in the House,” he said.

Wisconsin Gov. Scott Walker has warned against a government shutdown, while Mr. Christie told CBS for a segment scheduled to air Monday, “I don’t think you hear responsible Republican leaders advocating a shutdown of the government.”

Mr. Cruz has found some allies among his potential rivals. Indeed, Sens. Rand Paul of Kentucky and Marco Rubio of Florida gave Mr. Cruz periodic breathers during his filibuster and said they supported his strategy.

Mr. Paul was effectively returning a favor to Mr. Cruz, who helped Mr. Paul early this year during his 13-hour filibuster against the nomination of John O. Brennan as CIA director, threatening to talk until the administration clarified its position on the use of drones on American soil.