Non-trivial identities in finite-rank plactic monoids

Let ${\mathsf{plac}}_n$ denote the plactic monoid of rank $n$.

Finite-rank plactic identity conjecture. For each $n \geq 4$, there is a non-trivial identity satisfied by ${\mathsf{plac}}_n$.

The cases of ranks one through three have known identities, while the full-rank plactic monoid satisfies no non-trivial identity. The conjecture concerns whether every finite-rank plactic monoid of rank at least four satisfies some non-trivial identity.