Quasi-polynomial behavior for counts of maximal lattice slices

Let $P$ be a lattice $d$-polytope, and let $\ell \in [d]$. Define $\operatorname{LD}_P^{\ell}(k)$ to be the number of $\ell$-dimensional slices of $P$ that contain the most lattice points among all $\ell$-dimensional slices of $P$. Maximal-slice counting conjecture. There exists a positive integer $q$ such that

$$\operatorname{LD}_P^{\ell}:\mathbb{N}_{\geq q} \to \mathbb{N}$$

agrees with a quasi-polynomial function of degree $d-\ell$. The previously established two-dimensional diameter-segment case motivates this proposed extension; a proof would require new ideas because the higher-dimensional relation between lattice-point counts and normalized volume is more subtle.