What I call a graph here allows parallel edges. Is the following problem #P-hard:
OUTPUT: the number of matchings of $G$.
It is known that counting matchings on 2-3-regular (even planar) graphs is #P-hard: in this paper it corresponds to the problem denoted $\#[1,1,0]|[1,1,0,0]$, which is hard according to the table page 10, and in that one it corresponds to the problem $\#$2-3-RBP-$\lambda$-Matchings with $\lambda = 1$, and it is also shown to be #P-hard. In that last paper it is also shown that counting matchings on $3$-regular planar graphs is hard (see page 122, second paragraph). But what I need is 3-regular bipartite graphs.