Bijection between standard pairs and web basis indices

Let ${\mathcal{M}}$ be a positroid, and let an ordered pair $(I,J)$ of $k$-element subsets be ${\mathcal{M}}$-standard if $I=\pi_k(x)$ and $J=\pi_k(y)$ for some $u\leq x\leq y\leq v$, where $u\leq v$ is an interval in Bruhat order such that the Richardson subvariety $X^v_u$ projects birationally to the positroid variety $\Pi_{\mathcal{M}}$. Let ${\mathcal{A}}({\mathcal{M}})$ denote the indexing set of the basis constructed in the paper, and let $\theta$ be the map of Proposition. The standard-pair bijection conjecture. The map $\theta$ restricts to a bijection between ${\mathcal{M}}$-standard pairs $(I,J)$ and the set ${\mathcal{A}}({\mathcal{M}})$. This would identify the standard-monomial degree-two basis of ${\mathbb C}[\Pi_{\mathcal{M}}]$ with the basis indexed by ${\mathcal{A}}({\mathcal{M}})$, clarifying the relationship between standard monomials and the dimer/web construction. The supplied text gives no resolution status.