Gorenstein-injective realization conjecture for positroid categories

Let $v otin\quotmax$ and $w\geq v$, let $b1a\sigma=v^{-1}w$ be the resulting permutation decorated by declaring a fixed index $j$ black if $j\in v^{-1}[k]$, and set $I=v^{-1}[k]$. Let $B_{\mathcal{I}(\sigma)}=\bigoplus_i M_{I_i}$, and let $\gi(B_{\mathcal{I}(\sigma)})$ denote the category of Gorenstein-injective objects defined by $\operatorname{Ext}^1(B_{\mathcal{I}(\sigma)}^{\vee},M)=0$. With $\pi_I$ the functor to the positroid module category, the Gorenstein-injective realization conjecture.

$$\operatorname{add}\bigl(\pi_I(\gi(B_{\mathcal{I}(\sigma)}))\bigr)=\lec v w.$$

This predicts that the additive closure of the projected Gorenstein-injective category recovers the category associated with the interval from $v$ to $w$. The source presents this as an open conjecture connecting decorated permutations, necklaces, and categories attached to positroid varieties.