Is there an analogue of Frucht's theorem for sandpile groups? In other words, is it the case that for every abelian group $G$, there exists a graph $H$ such that the sandpile group of $H$ is isomorphic to $G$? If not, is the truth of falsity of this still an open question?

If $G=\bigoplus_{i=1}^N \mathbb{Z}_{k_i}$, then $G$ is the sandpile group of the (multi)graph on $N$ non-sink vertices where the $i$-th vertex is joined with the sink by $k_i$ edges.
