The polynomiality conjecture for pure-K commuting matrix varieties

Let $\alpha=(K(a_1),\dots,K(a_d))$ be a pure-$K$ leading term datum, and let $k$ be any field. Consider the affine variety

$$V(\alpha):=\{(X,Y)\}\subseteq\operatorname{Mat}_d(k)^2$$

defined by

$$X^2=Y^3,\qquad [X,Y]=0,$$

and by the vanishing conditions $X_{ij}=0$ if $\mu^0_j\prec T^3\mu^0_i$ and $Y_{ij}=0$ if $\mu^0_j\prec T^2\mu^0_i$. The pure-K commuting-matrix polynomiality conjecture. The motive $[V(\alpha)]$ in the Grothendieck ring $K_0(\operatorname{Var}_k)$ is a polynomial in $\mathbb{L}$.

This conjecture is presented as an independent route toward the motivic rationality question in the $q$-variable. Its status is open in the supplied text.