The plactic monoid cohomology exterior-algebra conjecture

Let $A$ be a finite ordered set, let $\Bbbk$ be a commutative unital ring, and let $\mathbf{Pl}_A$ denote the plactic monoid on $A$. Theorem~ gives an injective algebra map from the exterior algebra $\Lambda(\Bbbk A)$ into the Hochschild cohomology $H^*(\mathbf{Pl}_A;\Bbbk)$. Plactic cohomology conjecture. This injection is an algebra isomorphism:

$$\Lambda(\Bbbk A)\simeq H^*(\mathbf{Pl}_A;\Bbbk).$$

The conjecture asks whether the exterior subalgebra already accounts for all Hochschild cohomology of the plactic monoid; the corresponding injectivity result is proved in the paper, while surjectivity is left open.