“This just does not make any sense to me,” said Christine Todd Whitman, who served as head of the E.P.A. during the first George W. Bush administration. “Everybody breathes the same air, Democrats or Republicans. It does not matter. This is about keeping people healthy.”

But the Fitzgerald family has had influential allies. In addition to Mr. Pruitt, they include Representative Diane Black, a Republican who is a candidate for Tennessee governor, and Tennessee Technological University, a state university that produced a study minimizing pollution problems associated with the trucks.

Ms. Black introduced legislation in 2015 to protect the loophole when it was first in line to be eliminated by a stricter diesel emissions rule under the Obama administration. That bill failed, but after the election of Mr. Trump, she turned to Mr. Pruitt to carve out an exemption to the new rule — scheduled to take effect last month — and presented him with the study from Tennessee…

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