As Chairman Jason Chaffetz suggested Wednesday he may not finish his congressional term, top Republicans on the Oversight and Government Reform Committee mostly avoided weighing in on whether they would seek the seat under a GOP White House.

The Utah Republican unabashedly used the position of being the House’s top inquisitor to become nationally prominent in a coveted seat. But he did so under a Democratic administration that the House GOP despised.

[Chaffetz Lays out Different Direction for Oversight]

Now, the next leader of the powerful panel will have to deal with a White House that belongs to their own political party, which could complicate matters of accountability.

Republicans would face the conundrum of being put in a position that requires them to investigate an administration belonging to their own party, Brookings Institute fellow Molly Reynolds said.