Crystal-basis conjecture for the type-D module $V_\theta$

Let $I$ be the parameter set with involution $\theta$, and assume that no $i\in I$ satisfies $\theta(i)=i$. Let $V_\theta$ be the type-D module constructed in the source, with operators $E_i$ and $F_i$. Define modified root operators by

$$\widetilde E_i(u)=\sum_{n\geqslant1}F_i^{(n-1)}u_n,\qquad \widetilde F_i(u)=\sum_{n\geqslant0}F_i^{(n+1)}u_n$$

when $u=\sum_{n\geqslant0}F_i^{(n)}u_n$ with $E_iu_n=0$. Let $L_\theta$ be the $\mathbf A_0$-submodule generated by $\widetilde F_{i_1}\cdots\widetilde F_{i_\ell}\vac_\pm$, and let $B_\theta$ be the corresponding subset of $L_\theta/q_sL_\theta$. Crystal-basis conjecture. One has $\widetilde F_iL_\theta\subset L_\theta$ and $\widetilde E_iL_\theta\subset L_\theta$; $B_\theta$ is a basis of $L_\theta/q_sL_\theta$; and $\widetilde F_iB_\theta\subset B_\theta$ and $\widetilde E_iB_\theta\subset B_\theta\sqcup\{0\}$. This conjecture asserts that the explicitly generated lattice and crystal behave as a crystal basis for $V_\theta$ under the stated no-fixed-point condition.