One of those members, Rep. John Fleming of Louisiana, said he thought leadership would cool its efforts to push an immigration rewrite this year.

“My sense is that the consensus here is that we should not move forward and that leaders will abide by that,” Fleming told CQ Roll Call. “On a political basis, this is a suicide mission for Republicans. Why would we want to change the topic for a very toxic problem Democrats have with Obamacare?”

Fleming said Financial Services Chairman Jeb Hensarling of Texas, Science Chairman Lamar Smith of Texas, and Budget Vice Chairman Tom Price of Georgia — all of whom are influential in conservative circles — spoke in front of the conference to say an overhaul should not move forward this year.