The type-B LLT conjecture for the $p_1$-orbit

Let $I=\{p_1^n;n\in\mathbb Z_{\mathrm{odd}}\}$, assume that none of $\pm1$ and $\pm p_0$ lies in $I$, and let $\theta(a)=a^{-1}$. Let $\mathfrak g_I$ be the associated Lie algebra and let $V_\theta(\lambda)$ be the module of the preceding construction with $\lambda=0$. Let $\mathcal K_I$ be the Grothendieck group of the relevant affine Hecke-algebra representations, viewed inside the specialization of the module at $q=1$. The type-B LLT conjecture. The module $V_\theta(\lambda)$ has a crystal basis and a global basis, and the elements of $\mathcal K_I$ associated with irreducible representations correspond to the upper global basis of $V_\theta(\lambda)$ at $q=1$. This is the type-B analogue of the Lascoux–Leclerc–Thibon conjecture; the source notes that the assertion is already a theorem when $p_1$ is not a root of unity, but gives no general resolution status.