Vanishing and leading-term conjecture for generalized regularization q-decomposition numbers

Let $a$ and $b$ be positive integers. Let $\lambda$ and $\mu$ be partitions of the same size, with $\lambda$ $\operatorname{Reg}_{a,b}$-valid and $\mu$ $(a,b)$-regular. Write $d_{\lambda,\mu}(q)$ for the corresponding $q$-decomposition number, let $\trianglelefteq$ denote dominance order, and let $s(\lambda)$ be the number of $(a,b)$-steep hooks of $\lambda$ divisible by $b$. Generalized regularization q-decomposition conjecture. If $\lambda^{\operatorname{Reg}_{a,b}}\ntrianglelefteq\mu$, then

$$d_{\lambda,\mu}(q)=0.$$

Moreover,

$$d_{\lambda,\lambda^{\operatorname{Reg}_{a,b}}}(q)=q^{s(\lambda)}.$$

This conjecture predicts a dominance-order vanishing criterion and an explicit leading coefficient for the generalized regularization term in the $q$-decomposition matrix. It is motivated by the relationship between the Mullineux involution, generalized regularization, and crystal-basis $q$-decomposition numbers.