Minkowski's covering-radius conjecture for unimodular lattices

Let $X_d$ be the space of lattices under consideration, let $A$ be the group acting on $X_d$, and let $\mu=\operatorname{CovRad}_N$ be the covering radius associated with $N(\mathbf{v})=\prod_{i=1}^d|v_i|$. Minkowski's conjecture. For every $\Lambda\in X_d$,

$$\mu(\Lambda)\leq 2^{-d}=\mu(\mathbb{Z}^d)$$

and equality holds if and only if $\Lambda\in A\mathbb{Z}^d$. This conjecture has been proved for $d\leq 10$; its general status is not established in the supplied text.