The non-integer eigenvalue conjecture for non-triangular RSK operators

Let $(\sigma,\pi)$ be a reduced pair of compositions, and let ${\sf RSK}_{\sigma,\pi}$ be the associated RSK operator. A pair is triangular in the sense used in the paper. Non-integer eigenvalue conjecture. If $(\sigma,\pi)$ is not triangular, then ${\sf RSK}_{\sigma,\pi}$ has a non-integer eigenvalue. Triangular pairs are shown to have only the eigenvalues $1$ and $-1$, while the conjecture for non-triangular reduced pairs remains open.