Let $K$,$L$,$M$ be convex lattice polytopes (so their vertices are in $\mathbb{Z}^n$) in $\mathbb{R}^n$ satisfying $K+L\subseteq M+M$ (Minkowski sum). Do we always have $$|K\cap\mathbb{Z}^n|\cdot|L\cap\mathbb{Z}^n|\le |M\cap\mathbb{Z}^n|^2 ?$$

I looked at books/papers on Erhart polynomial and Brunn-Minkowski inequality etc., but did not find an satisfying answer. Any comment? Thanks.