Finite-dimensional quotient conjecture for exceptional Verma modules

Let $n\in\mathbb{Z}$, and let $\mathbb{M}(r+\kappa,r,[\iota;n],\zeta)$ be an exceptional Verma module. Suppose $j$ satisfies $0\neq j\neq\kappa-1$, and define the condition

$$\zeta=\iota\left(q^{n-j}[\kappa-j]+q^{j-\kappa-n}[j]\right).$$

Finite-dimensional quotient conjecture. The module $\mathbb{M}(r+\kappa,r,[\iota;n],\zeta)$ has a finite-dimensional irreducible quotient $L$ if and only if this condition holds. In that case, the $U_q(\mathfrak{sl}_2)$-character of $L$ is

$$[\kappa+1]+[\kappa-1]+\cdots+[\kappa+1-2\min\{j,\kappa-j\}].$$

The conjecture is motivated by explicit examples and by an argument showing that exceptional Verma modules admitting finite-dimensional quotients must satisfy the displayed condition; the sufficiency and the asserted character formula remain open.