Eigenvalue-determination conjecture for sandpile groups

Let $M$ be a generating multiset for a Cayley graph, and suppose the greatest common divisor of all generator multiplicities in $M$ is $1$. Eigenvalue-determination conjecture. The sandpile group depends only on the set of eigenvalues generated by $M$, and not on the specific choice of $M$.

This predicts that, under the stated primitivity condition on generator multiplicities, the spectral data determines the sandpile group. The source presents it as a remaining conjecture based on computational data.