Geometric polynomial conjecture for lower Bruhat interval sizes

Let $\Phi$ be an affine Weyl group of rank $n$, let $a\in\mathcal{F}$, and let $q_a$ be the map sending a dominant coweight $\lambda$ to the cardinality of its lower Bruhat interval, $q_a(\lambda)=|\mathcal{I}_a(\lambda)|$. A geometric polynomial is a rational combination of volume polynomials of faces of permutohedra.

Geometric polynomial conjecture. For any affine Weyl group, the map $q_a$ is a geometric polynomial.

The paper reports computations showing that the quasi-polynomial supplied by the lattice-point argument is in fact polynomial, and that this polynomial is geometric. The conjecture predicts this phenomenon for every affine Weyl group.