Crystal and global-basis conjecture for type-D affine Hecke representations

Let $I=\{p^n\mathbin{;}n\in\mathbb Z_{\mathrm{odd}}\}$, with neither $1$ nor $-1$ in $I$, and let $\theta(a)=a^{-1}$. Let $K_{D,I}$ be the Grothendieck group of finite-dimensional type-$I$ modules over the type-D affine Hecke algebras, and let $(V_\theta)_\mathbf A$ be the integral form of the module $V_\theta$. Type-D representation-theoretic conjecture. The following hold: $K_{D,I}\simeq (V_\theta)_\mathbf A/(q-1)(V_\theta)_\mathbf A$; $V_\theta$ has a crystal basis and an upper global basis; the classes associated with irreducible representations correspond to the upper global basis at $q=1$; and the operators $\widetilde F_i$ and $\widetilde E_i$ correspond respectively to the representation-theoretic operators $\widetilde f_i$ and $\widetilde e_i$. This would realize the Grothendieck group of type-D affine Hecke representations through the crystal and global-basis structure of $V_\theta$.