Character conjecture for dual canonical basis elements and integral Hecke superalgebra modules

Let $r\geqslant 2$, let ${\cal H}_m(t)$ be the relevant Hecke-Clifford superalgebra, and let ${\cal C}_{m,r}$ be its category of integral modules whose eigenvalues lie in $\{t(1),\ldots,t(r)\}$. Let ${\cal R}_{m,r}$ be the complexified Grothendieck group of ${\cal C}_{m,r}$, and let $\Phi$ be the quantum-shuffle realization of $U_q(\mathfrak n)$. For $b^*\in{\bf B}^*$ of principal degree $m$, specialize the coefficients of $\Phi(b^*)$ at $q=1$. Character conjecture. The specialized element is the character of an irreducible integral ${\cal H}_m(t)$-module. This would identify the dual canonical basis with the simple-module basis in the character realization of the Grothendieck ring. The preceding discussion gives the algebraic and crystal-theoretic correspondence, but the asserted character realization is not established in the supplied text.