Weighted Levi–Hadwiger covering conjecture

Let $K\subseteq\mathbb{R}^{n}$ be a convex body, and let $N_{\omega}(K,T)$ denote its weighted covering number. Weighted Levi–Hadwiger covering conjecture.

$$\lim_{\lambda\to1^{-}}N_{\omega}(K,\lambda K)\le 2^{n}.$$

Moreover, equality holds if and only if $K$ is a parallelotope. The source proves the bound for centrally symmetric convex bodies and proves the equality characterization in that case; the assertion for arbitrary convex bodies remains open.