The support approximation conjecture for vexillary involutions

Let $I^{\mathsf{vex}}_\infty$ be the set of vexillary involutions, and let $\mathcal{B}_{\mathsf{inv}}(z)$, $\mathcal{B}_{\mathsf{inv}}^+(z)$, and $\operatorname{supp}(\operatorname{GC}^{\mathsf{O}}_z)$ denote the sets defined in the paper. Support approximation conjecture. If $z \in I^{\mathsf{vex}}_\infty$, then

$$\mathcal{B}_{\mathsf{inv}}(z)\subseteq \operatorname{supp}(\operatorname{GC}^{\mathsf{O}}_z) \subseteq \mathcal{B}_{\mathsf{inv}}^+(z).$$

The conjecture is motivated by computational verification for all vexillary $z\in I_{11}$ and extends the preceding containment results; the general statement remains open.