The disjoint-cone conjecture for SL₂-plethysm generating functions

The bivariate generating function is

$$A_\mu(z,q) = \sum_{k\geqslant 1, h\geqslant 0} a_{\mu[h]}^{[k]} q^k z^h.$$

A weighted lattice-point cone conjecture. The generating function $A_\mu(z,q)$ is a weighted generating function of lattice points of a disjoint union of convex rational half-open cones.

Positive expansions of $A_\mu(z,q)$ into sums of $q$-Ehrhart series yield such interpretations for the computed examples, but Kahle and Michałek showed that this conjecture is false because it would violate Ehrhart reciprocity for closed polytopes.