Keep My Star From Rising, Please

From the Department of Up Is Down:

GOPers inside the Beltway and around the country discussed a list of
about 20 high-profile or rising-star Republicans being circulated by
Duncan opponents who want a bigger name at the RNC.

The list contained such GOP luminaries as former Speaker Newt Gingirch (Ga.), former Florida Gov. Jeb Bush, Missouri Gov. Matt Blunt...

Is it possible to be forced from running for a second term as governor due to rampant unpopularity and still be categorized as a "rising star"?