WASHINGTON — The contest for the No. 3 spot in the House GOP has turned into conservatives’ last, best shot at joining the congressional leadership after getting shut out of the two top jobs in the shake-up that followed majority leader Eric Cantor’s surprise primary defeat.

It’s become an intense intramural clash with no certain outcome, as two candidates from different ideological outposts and regions of the country — a conservative Southerner and an establishment-aligned Midwesterner — are challenged by a third who could play the role of spoiler for tea party hopes.

All three — Reps. Steve Scalise of Louisiana, Peter Roskam of Illinois and Marlin Stutzman of Indiana — were to make their case to rank-and-file lawmakers Wednesday ahead of a Thursday vote.

The job they’re vying for is majority whip, likely to become vacant because its current occupant, Rep. Kevin McCarthy of California, is the strong favorite to become the new majority leader in a separate vote.