The non-real eigenvalue conjecture for RSK operators

Let $(\sigma,\pi)$ be a pair of compositions, let $\ell(\sigma)$ and $\ell(\pi)$ denote their lengths, and let ${\sf RSK}_{\sigma,\pi}$ be the associated RSK operator. Non-real eigenvalue conjecture. If

$$\ell(\sigma),\ell(\pi)\geq 3,$$

then ${\sf RSK}_{\sigma,\pi}$ has a non-real eigenvalue. The conjecture has been checked for $\sigma,\pi\in {\mathbb N}^3$ of degree $d\leq 8$, but remains open in general.