The polynomial-count conjecture for double-affine transverse slices

Let $W_{\mathcal{T}}$ be the double-affine Weyl monoid, with length function $\ell$, and let $k$ be a finite field of cardinality $q$. For $x\leq y$, consider the group-theoretic transverse slice

$$I x\cdot I/I\cap I_\infty y\cdot I/I.$$

The transverse-slice counting conjecture. There exists a polynomial $R_{x,y}\in\mathbb{Z}[v]$, independent of $k$, of degree $\ell(y)-\ell(x)$ such that

$$\#\bigl(I x\cdot I/I\cap I_\infty y\cdot I/I\bigr)=R_{x,y}(q).$$

This conjecture is intended to express that the transverse slices have dimension $\ell(y)-\ell(x)$ and to provide a group-theoretic definition of the length function. The paper does not establish the required polynomial-counting or degree statement, so it remains open.