Support description conjecture for almost vexillary permutations

Let $w$ be an almost vexillary permutation, let $D(w)$ be its associated diagram, and let $\mathcal{SBD}(D(w),\emptyset,A)$ be the stated set of combinatorial objects for a subset $A\subset D(w)$. For such an object $\mathcal D$, write $\mathrm{wt}(\mathcal D)$ for its weight, and let $\mathrm{supp}(\mathfrak G_w)$ denote the support of the Grothendieck polynomial $\mathfrak G_w$.

Almost vexillary support conjecture. For every almost vexillary permutation $w$, there exists a subset $A\subset D(w)$ such that

$$\mathrm{supp}(\mathfrak G_w)=\{\mathrm{wt}(\mathcal D)\colon \mathcal D\in\mathcal{SBD}(D(w),\emptyset,A)\}.$$

The supplied context does not identify this assertion as proved or refuted, so it is recorded as open. The notation suggests a combinatorial model for the support, but the excerpt does not provide its definitions.