Minkowski's uniqueness conjecture for the multiplicative covering radius

Let $d\geq 2$, let $G=\operatorname{SL}_d(\mathbb{R})$, let $\Gamma=\operatorname{SL}_d(\mathbb{Z})$, and let $X_d=G/\Gamma$, identified with the space of unimodular lattices in $\mathbb{R}^d$. Let $A$ be the group of diagonal matrices in $G$ with determinant $1$. For a lattice $x\in X_d$, define the multiplicative covering radius $\mu(x)=\operatorname{CovRad}_N(x)$ using $N((v_1,\ldots,v_d))=\prod_{i=1}^d\lvert v_i\rvert$. Minkowski's conjecture. For every $d\geq 2$ and every $x\in X_d$,

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

and

$$\mu(x)=2^{-d}\quad\Longleftrightarrow\quad x\in A\mathbb{Z}^d.$$

Minkowski's conjecture concerns the sharp upper bound and the uniqueness of its maximizer for the multiplicative covering radius. The paper's abstract states that it constructs infinitely many counterexamples in positive characteristic, while this real-characteristic formulation is the classical conjecture attributed to Minkowski.