Demazure crystal decomposition conjecture for symplectic bounded reduced factorizations

Let $I^{\mathsf{fpf}}_\infty$ be the set of finite fixed-point-free involutions, let $z\in I^{\mathsf{fpf}}_\infty$ have no fpf-descents greater than $n$, and let $\phi$ be any flag. A Demazure $\mathfrak q_n$-crystal is a $\mathfrak q_n$-crystal isomorphic to $\mathrm{BRF}^{\mathsf{Sp}}_n(\alpha)$ for a skew-symmetric weak composition $\alpha\in\mathbb N^n$.

Symplectic Demazure crystal conjecture. The crystal $\mathrm{BRF}^{\mathsf{Sp}}_n(z,\phi)$ is a direct sum of Demazure $\mathfrak q_n$-crystals.

This is a crystal-theoretic generalization of the symplectic part of the involution Schubert polynomial expansion conjecture; taking characters with the standard flag yields the polynomial statement. The paper identifies it as an open problem.