Naive dominance conjecture for polynomial modules of quantum affine superalgebras

Let $\Psi$ be a monomial in $Y_{i,a}$ for $1\leq i\leq M$ and $a\in\mathbb{C}^{\times}$, so that $L(\Psi)$ is a polynomial module. Assume that every $\ell$-weight of $L(\Psi)$ other than the highest one is not a monomial in $Y_{i,a}$, $\widetilde{Y}_{j,a}$ and $D^{\pm1}$ with $i\leq M<j$ and $a\in\mathbb{C}^{\times}$. Naive dominance conjecture. Under this assumption, the algorithm is well-defined. This conjecture concerns a naive definition of dominance for polynomial modules and their duals; it is supported by a number of examples, while the supplied text gives no resolution.