Symplectic reduced-tableau Demazure crystal conjecture

Let $I^{\mathsf{fpf}}_\infty$ be the set of finite fixed-point-free involutions, and let $z\in I^{\mathsf{fpf}}_\infty$ have no fpf-descents greater than $n$. Let $T$ be an $\mathsf{Sp}$-reduced tableau for $z$, and let $\mathrm{BRF}^{\mathsf{Sp}}_n(T)$ denote its bounded reduced-factorization crystal. A skew-symmetric weak composition is a weak composition $\alpha\in\mathbb N^n$ satisfying the skew-symmetry condition used in the paper.

Symplectic reduced-tableau conjecture. There is a $\mathfrak q_n$-crystal isomorphism

$$\mathrm{BRF}^{\mathsf{Sp}}_{n}(T)\cong \mathrm{BRF}^{\mathsf{Sp}}_n(\alpha)$$

for some skew-symmetric weak composition $\alpha=\alpha^{\mathsf{Sp}}(T)\in\mathbb N^n$.

The source states this as an equivalent reformulation of the symplectic Demazure crystal conjecture. It remains open.