Uyemura-Reyes's integrality conjecture for random-to-random shuffling

Let $\upsilon_n$ be the group-algebra element that is $n^2$ times the transition matrix for random-to-random shuffling of a deck of $n$ cards.

Uyemura-Reyes's conjecture. The eigenvalues of $\upsilon_n$ are rational integers.

This conjecture concerns the spectrum of the normalized random-to-random shuffle operator and is motivated by computations of its nullspace and representation-theoretic structure. The paper uses the conjecture to imply integrality of the spectrum of the Laplacian associated with the complex of injective words, but the supplied text gives no resolution of the conjecture.