Vexillary involution crystal isomorphism conjecture

Let $I_\infty$ be the set of finite involutions, let $z\in I_\infty$ be vexillary, meaning $2143$-avoiding, and let $c(z)$ be its code, which is a symmetric weak composition. Assume that the largest descent of $z$ is at most $n$.

Vexillary involution crystal conjecture. There is an isomorphism of $\mathfrak q^+_n$-crystals

$$\mathrm{BRF}^{\mathsf{O}}_n(z)\cong \mathrm{BRF}^{\mathsf{O}}_n(c(z)).$$

This is a crystal-theoretic strengthening of the conjectural identity $\mathfrak S^{\mathsf{O}}_z=\kappa^{\mathsf{Q}}_{c(z)}$ for vexillary involutions. The source reports supporting computations, but the conjecture is unresolved.