Betke-Henk-Wills conjecture on lattice point enumerators

Let $\mathcal{K}^d_0$ be the set of $o$-symmetric convex bodies in $\mathbb{R}^d$, let $\mathcal{L}^d$ be the space of full-rank lattices, and for $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$ let

$$G(K,\Lambda)=|K\cap\Lambda|,$$

with successive minima

$$\lambda_i(K,\Lambda)=\inf\{\lambda>0:\dim(\lambda K\cap\Lambda)\geq i\},\qquad 1\leq i\leq d.$$

Betke-Henk-Wills conjecture. For any $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$,

$$G(K,\Lambda)\leq\prod_{i=1}^d\left\lfloor\frac{2}{\lambda_i(K,\Lambda)}+1\right\rfloor.$$

The conjecture is a discrete analogue of Minkowski's Second Theorem, seeking to bound the global lattice point count using the successive minima. It is established for orthogonal parallelotopes but remains unresolved for $d\geq5$ and general convex bodies.