The vexillary Grothendieck–Schubert character conjecture

Let $w\in S_n$ be a vexillary permutation. Write $\mathfrak G_w^\mathrm{top}$ for the top-degree homogeneous component of its Grothendieck polynomial, and let $D^\mathrm{top}(w)$ be the associated diagram. Denote by $\chi_D$ the character associated with a diagram $D$. Vexillary Grothendieck–Schubert character conjecture. If $w\in S_n$ is vexillary, then $\mathfrak G_w^\mathrm{top}$ is an integer multiple of $\chi_{D^\mathrm{top}(w)}$. The statement strengthens the preceding result relating top homogeneous components of Grothendieck polynomials to diagram characters; it was tested for all vexillary permutations $w\in S_n$ with $n\leq 9$, but remains unproved in the supplied text.