Is there a known bijective proof of Ramanujan's congruence for the partition function modulo 5? E.g., is there a construction that for every $n$ congruent to 4 mod 5 gives a permutation of the partitions of $n$ that increases the Dyson rank of each partition by 1 mod 5?
It's hard to be absolutely certain of course, but I would say no. The most recent reference I can find mentioning the absence of such a proof is this article by Bessenrodt and Pak (2003). I also got some negative answers from modular form specialists to whom I asked the very same question fairly recently... 

