GL-equivalence conjecture for sandpile groups of Cayley graphs

Let two Cayley graphs have generator multiplicities, and let $GL$-equivalence mean equivalence under the natural action of the relevant general linear group on the generators. GL-equivalence conjecture. The two Cayley graphs have the same sandpile group if and only if their generator multiplicities are the same up to $GL$-equivalence.

This proposes a classification of the sandpile group by the linear-equivalence class of the generator multiplicities. The source lists it among remaining conjectures based on data.