Subjects or conjectures with infinite hanging fruit?
Is there any area of study or conjecture in research mathematics where it is either proven or highly suspected that a unique proof is required for an infinite (or extremely large) number of cases? For example, something where “proven for all dimensions” is known to not be possible.
By hanging fruit I don’t mean the proof has to be easy, in fact infinitely many difficult proofs is more of what I’m curious about.