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

Assume that no $i\in I$ satisfies $\theta(i)=i$, and use the lattice $L_\theta$, its bar-conjugate $L_\theta^-$, and the integral module $(V_\theta)_\mathbf A$ defined from the bar operator and the divided-power algebra. Global-crystal-basis conjecture. The triple

$$(L_\theta,L_\theta^-,(V_\theta)_\mathbf A)$$

is balanced. Equivalently, the associated intersection maps isomorphically onto $L_\theta/q_sL_\theta$, so the lifts of the crystal basis form an upper global basis of $V_\theta$. This is the global refinement of the preceding crystal-basis conjecture and would identify canonical basis elements for the type-D representation-theoretic model.