Why do pedestal matrices have integer eigenvalues?

The pedestals were introduced in order to understand the MacMahon formula. It enumerates plane partitions (3D Young diagrams), and pedestals allow one to extend the formula to higher dimensions. Pedestals can naturally be arranged into square matrices, with entries being certain monomials. It was observed experimentally that the eigenvalues of any such matrix are polynomials (!) in the corresponding variables, with integer coefficients. After many years this phenomenon was explained in the paper "The miracle of integer eigenvalues, by Richard Kenyon, Maxim Konsevich, Oleg Ogievetsky, Andrei Pohoata, Will Sawin, and myself.