Cancellation-free identity for graded characters of level-zero Demazure submodules in type C

Let $W$ be the Weyl group of type $C_n$, let $P^+$ be the set of dominant weights, and let $\varepsilon_k$, $\mathcal{A}$, $\Theta_k$, $\Gamma_k(k)$, $\operatorname{gch} V_x^-(\lambda)$, $\operatorname{end}(B)$, $\operatorname{down}(B)$, and $\mathbf{p}_{\overline{m},k}(w)$ have the meanings specified in the paper. For $x=wt_\xi\in W_{\mathrm{af}}$ with $w\in W$ and $\xi\in Q^\vee$, take $m=1,\ldots,n$ and $\lambda\in P^+$ such that $\lambda+\varepsilon_k\in P^+$ for all $k=1,\ldots,n$ and $\lambda-\varepsilon_k\in P^+$ for $k=m+1,\ldots,n$.

Cancellation-free identity. There exists some $m\leq l\leq n$ such that

$$\begin{aligned} e^{-w\varepsilon_m}\operatorname{gch}V_x^-(\lambda) ={}&q^{-\langle\varepsilon_m,\xi\rangle}\sum_{B\in\mathcal{A}(w,\Theta_m)}(-1)^{|B|}\operatorname{gch}V_{\operatorname{end}(B)t_{\operatorname{down}(B)+\xi}}^-(\lambda-\varepsilon_m)\\ &+\sum_{k=m+1}^n q^{-\langle\varepsilon_k,\operatorname{down}(\mathbf{p}_{\overline m,\overline k}(w))+\xi\rangle}\\ &\quad\times\sum_{B\in\mathcal{A}(\operatorname{end}(\mathbf{p}_{\overline m,\overline k}(w)),\Theta_k)}(-1)^{|B|}\operatorname{gch}V_{\operatorname{end}(B)t_{\operatorname{down}(B)+\operatorname{down}(\mathbf{p}_{\overline m,\overline k}(w))+\xi}}^-(\lambda-\varepsilon_k)\\ &+\sum_{k=1}^l q^{\langle\varepsilon_k,\operatorname{down}(\mathbf{p}_{\overline m,k}(w))+\xi\rangle}\\ &\quad\times\sum_{B\in\mathcal{A}(\operatorname{end}(\mathbf{p}_{\overline m,k}(w)),\Gamma_k(k))}(-1)^{|B|}\operatorname{gch}V_{\operatorname{end}(B)t_{\operatorname{down}(B)+\operatorname{down}(\mathbf{p}_{\overline m,k}(w))+\xi}}^-(\lambda+\varepsilon_k). \end{aligned}$$

This is proposed as a cancellation-free inverse Chevalley-type identity for graded characters of level-zero Demazure submodules over quantum affine algebras of type $C$. The supplied text does not indicate whether the conjecture has been proved or disproved.