Natural algebra action on matchings for alternating Kasteleyn matrices

Let $A$ be a nonsingular alternating matrix, and let $D$ be the matrix-algebra deformation of $\mathbb{C}[\operatorname{coker} A]$ defined by the twisted relations in the paper. Let $G$ be a planar graph with nonempty matching set $P$, and suppose that $A$ is a Kasteleyn matrix of $G$. Kasteleyn algebra-action conjecture. There is a natural action of $D$ on $\mathbb{C}[P]$, possibly depending on how $G$ is embedded in the plane. Such an action would provide the algebraic structure underlying the proposed quantum-bijection interpretation of perfect matchings. The source gives no proof or resolution, so the conjecture remains open.