The covering conjecture for well-rounded unimodular lattices

Let $L$ be a well-rounded unimodular lattice in $\mathbb{R}^d$, equipped with the standard norm $|\cdot|$. Its covering radius is

$$\sup_{x \in \mathbb{R}^d} \inf_{y \in L} |x-y|.$$

Covering conjecture. The covering radius satisfies

$$\sup_{x \in \mathbb{R}^d} \inf_{y \in L} |x-y| \leq \frac{\sqrt{d}}{2}.$$

Equality holds if and only if $L=g\cdot\mathbb{Z}^d$ for some $g\in\mathrm{SO}_d(\mathbb{R})$. This conjecture gives a sharp geometric bound relevant to controlling the dimension of non-negative integral matrices with prescribed spectral radius. Its resolution status is not specified in the source.