The most obvious question is whether or not a permutation fair 5d30 set exists. If such a set exists, it would make a beautiful successor to the 4d12 set that is commercially available. If such a set does not exist, it would be nice to have a proof of that (whether a mathematical proof or via exhaustive computer search).

It's even possible that a six player permutation fair set of all 30-sided dice might exist. This is definitely the “Holy Grail” of the Go First Dice research.

What are the “smallest” possible sets for five or six players? “Smallness” of a set can be measured in a few different ways, and all current records are list on this page, but for more than four players, lowest bounds have not yet been proven.