I've come across this paper by Otmar Ertl in Github. If I understand it correctly, he points out that when you use p + q bit hashes, you put integers at most of value q into 2^p registers. Instead, he stores a q long vector and keeps counts in them so that the counts sum to 2^p.
Was this result known previously? Wikipedia does not mention this alternative for the HyperLogLog data structure. It compresses the data structure enormously.