Truncated $q$-character relation for asymptotic representations

Let $L(m)$ be a simple finite-dimensional representation whose dominant monomial satisfies $m\in\mathbb{Z}[Y_{i,aq^r}]_{(i,r)\in W,\ r\leq R}$ for $R\in\mathbb{Z}$. Let $\chi_q^{\leq R}(L(m))$ be its truncated $q$-character, and let $\chi_\ell^{\leq R}(L(m))$ be obtained by replacing each $Y_{i,a}$ by $[\omega_i]\ell_{i,aq_i^{-1}}\ell_{i,aq_i}^{-1}$. Define

$$W_R=\{(i,r)\in W\mid R\geq r>R-2d_i\},$$

and let $u_{i,r}$ be the maximum of $0$ and the powers of $Y_{i,q^r}$ occurring in monomials of $\chi_q^{\leq R}(L(m))$. Set

$$\boldsymbol{\Psi}_R=\prod_{(i,r)\in W_R}\boldsymbol{\Psi}_{i,q^{r+d_i}}^{u_{i,r}},\qquad \boldsymbol{\Psi}=m\boldsymbol{\Psi}_R.$$

Then $L(\boldsymbol{\Psi})$ and $L(\boldsymbol{\Psi}_R)$ belong to $\mathcal{O}^+$.

Truncated $q$-character relation. In $\operatorname{Frac}(K_0(\mathcal{O}^+))$,

$$\chi_\ell(L(\boldsymbol{\Psi}))=\chi_\ell^{\leq R}(L(m))\prod_{(i,r)\in W_R}\ell_{i,q^{r+d_i}}^{u_{i,r}}=\chi_\ell^{\leq R}(L(m))\chi_\ell(L(\boldsymbol{\Psi}_R)).$$

This is presented as an additional conjectural relation after results on tensor factorization and truncated characters; the source does not provide evidence of resolution.