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Mr. Rubio’s spokesman, Alex Conant, said the senator wants to give his Republican colleagues in that chamber space to work.

“We have a badly broken immigration system that is in dire need of reform, and Sen. Rubio worked hard to produce and pass the best proposal possible in a Democrat-controlled Senate,” Mr. Conant said. “The Senate’s work has created an important opportunity for the House to advance reform further, but they should be given deference to decide their own way forward.”