As Thursday's House vote on the bill to repeal the Affordable Care Act looms, the pressure is ramping up on GOP lawmakers who remain undecided or opposed to the legislation. Even after threats to their careers, invitations to the White House, special carve-outs for their states, amendments on their pet issues, and other tactics, critics of the bill still say enough members are holding strong to ensure the bill will fail on the House floor.

As the clock ticked down on Wednesday, Republican leaders made a new promise to the dissenters: that the Senate will add a provision gutting Obamacare's Essential Health Benefits (EHB) rule once the House passes the bill and sends it their way.

The EHB rule, which the current House GOP repeal bill retains, requires that insurance plans have to cover a basic minimum of health care services, including emergency room visits, hospitalization, outpatient services, maternity care, mental health and substance abuse services, prescription drugs, rehabilitative and habilitative services, lab tests, preventive care like vaccines, and vision and dental care for children.

This new promise was already winning over many previously dissenting conservatives, said Hudson (R-NC), who is part of the team charged with securing the votes for passage—enough that he's confident the bill will pass the House Thursday night.