The truncated q-character formula for real simple modules

Let $\xi$ be a height function and let $\ell\geq 2$. Let $\mathscr{C}^{\leq \xi}_\ell$ be the corresponding module subcategory, and let $L(m)\in\mathscr{C}^{\leq \xi}_\ell$ be a real simple module with no tensor factor $f_{i,r}$ for any $(i,r)\in\widehat{I}^{\leq \xi}_\ell\setminus\widehat{I}^{\leq \xi}_{\ell-1}$. Let $K^{\leq \xi}_\ell(m)$ be the general-kernel $A^{\leq \xi}_\ell$-module defined from the Laurent monomial $m=\prod_{(i,r)\in\widehat{I}^{\leq \xi}_\ell}z_{i,r}^{d_{i,r}}$, and let $F_{K^{\leq \xi}_\ell(m)}$ be its $F$-polynomial. Truncated q-character conjecture. Then

$$\widetilde{\chi}_{q}(L(m))_{\leq \xi}=F_{K^{\leq \xi}_\ell(m)}((\widehat{y}_{i,r})_{(i,r)\in\widehat{I}^{\leq \xi}_{\ell-1}}).$$

This conjectural formula would compute truncated q-characters from $F$-polynomials and thereby connect real simple quantum affine modules with cluster-category representations; the source does not provide a resolution.