Non-integer eigenvalues for non-triangular reduced pairs

Show that for any reduced weight pair (ω, τ) that is not triangular (i.e., not of the form l(ω) = 2 with ω1 = τ1), the matrix RSK_{ω,τ} possesses an eigenvalue that is not an integer.

Background

Triangular weights (defined by l(ω) = 2 and ω1 = τ1) yield upper-triangular matrices with eigenvalues confined to {±1}. For many other reduced weights, computations reveal non-rational eigenvalues, suggesting that integer spectra are exceptional. This conjecture aims to characterize precisely when non-integer eigenvalues must appear.