Lattice polytope face-count conjecture for dense faces

Let $S$ be a finite subset of $\mathbb{Z}^d$ with $d\geq 2$, and let $t$ be an integer satisfying

$$1\leq t\leq V^{1/d},$$

where $V=\operatorname{Vol}(\operatorname{conv}(S))>0$. A face of the lattice polytope $\operatorname{conv}(S)$ is $t$-dense if it contains more than $t$ points of $\mathbb{Z}^d$.

Lattice polytope face-count conjecture. The number of $t$-dense faces of $\operatorname{conv}(S)$, counting faces of every dimension, is

$$\mathcal{O}\left(\left(\frac{V}{t^d}\right)^{\frac{d-1}{d+1}}\right).$$

This conjecture generalizes the cited lemma and would provide a lower-bound framework for questions about large convex subsets in density-restricted point sets. Its resolution is not supplied in the source.