Convex uniform 5-polytopes
A reasonably complete discussion, with a list of all known examples, is at Uniform_5-polytope. The known examples form two infinite classes and 105 additional examples. All the known examples but one are "Wythoffian", and any unknown examples must be non-Wythoffian. The proposer believes that a talented undergraduate could put this to rest.
* indicates original appearance(s) of problem.