The classification conjecture for compatible algebras with straightening laws on a disjoint union of chains

Let $P$ be a finite poset that is a disjoint union of chains. Write ${\mathcal J}(P)$ for the poset of order ideals of $P$, and let $K[\mathcal O(P)]$, $K[\mathcal C(P^)]$, and $K[\mathcal C(P^{*})]$ denote the corresponding toric rings, regarded as compatible algebras with straightening laws on ${\mathcal J}(P)$ over $K$. Classification conjecture. If $P$ is a disjoint union of chains, then the compatible algebras with straightening laws on ${\mathcal J}(P)$ over $K$ are precisely $K[\mathcal O(P)]$, $K[\mathcal C(P^)]$, and $K[\mathcal C(P^{*})]$. The question asks for a classification of all compatible algebras with straightening laws on ${\mathcal J}(P)$; the stated claim gives the proposed complete list in the case where $P$ is a disjoint union of chains. No resolution status is provided in the source.