Non-real eigenvalues when both weights have length at least three

Determine whether, for any weight vectors ω ∈ N^m and τ ∈ N^n with lengths l(ω) ≥ 3 and l(τ) ≥ 3 and equal sums |ω| = |τ|, the matrix RSK_{ω,τ}—the restriction of the Robinson–Schensted–Knuth (RSK) linear operator to the weight space R_{ω,τ}—has at least one non-real complex eigenvalue.

Background

Section 5 studies spectral properties of the matrices RSK_{ω,τ}. It is shown that the only rational eigenvalues are ±1, and that for triangular weights all eigenvalues are ±1. Motivated by extensive computations, the authors conjecture that when both weight lengths are at least three, some eigenvalue is non-real, indicating genuinely complex spectral behavior beyond the special families where spectra are integers.