Generalized lattice-point bound for families of convex bodies

Let $K_1,\dotsc,K_n\subset\mathbb{R}^d$ be convex bodies and let $\Lambda$ be a lattice. Let $\mathbf{e}_1,\dotsc,\mathbf{e}_d$ be a basis of $\Lambda$, and let $\Lambda^i$ be the $\mathbb{Z}$-span of $0,\mathbf{e}_1,\dotsc,\mathbf{e}_i$. Let $q_1\geqslant q_2\geqslant\dotsb\geqslant q_d\geqslant q_{d+1}$ be positive integers satisfying

$$\mathfrak{D}K_j\cap q_i(\Lambda\setminus\Lambda^{i-1})=\varnothing$$

for all $1\leqslant j\leqslant n$ and $1\leqslant i\leqslant d$, and

$$(K_j-K_l)\cap q_{d+1}\Lambda=\varnothing$$

for all $1\leqslant j,l\leqslant n$ with $j\neq l$. Here $\mathfrak{D}K_j=K_j-K_j$ and $G(K_j,\Lambda)$ denotes the lattice-point enumerator used in the source.

Generalized lattice-point bound conjecture. Under these conditions,

$$\sum_{j=1}^n G(K_j,\Lambda)\leqslant\prod_{i=1}^d q_i.$$

The conjecture is presented as a more general form of the Betke–Henk–Wills inequality. The source does not provide a resolution of this general statement; it proves the relevant bound for ellipsoids and discusses weaker problems that would imply the desired inequality.