Positive-characteristic upper-bound conjecture for the Minkowski spectrum

Fix an integer $d\geq 2$ and a prime power $q$. Let $\left[\mathcal{L}_d\right]$ be the space of homothety classes of lattices, let $\tilde{\mathcal{K}}^d$ be the ambient vector space, and define $N(\mathbf{v})=\prod_{i=1}^d\lvert v_i\rvert$. For $\mathfrak{x}\in\left[\mathcal{L}_d\right]$, define its Minkowski function by

$$\mu(\mathfrak{x})=\operatorname{CovRad}_N(\mathfrak{x})=\frac{1}{\lvert\det(g)\rvert}\sup_{\mathbf{v}\in\tilde{\mathcal{K}}^d}\inf_{\mathbf{u}\in\mathfrak{x}}N(\mathbf{v}-\mathbf{u}),$$

where $g\mathcal{R}^d$ represents the homothety class of $\mathfrak{x}$. Positive-characteristic Minkowski conjecture. For every $\mathfrak{x}\in\left[\mathcal{L}_d\right]$,

$$\mu(\mathfrak{x})\leq q^{-d}=\mu\left(\left[\mathcal{R}^d\right]\right).$$

This provides the conjectured upper bound for the positive-characteristic Minkowski spectrum. The paper's abstract states that it constructs infinitely many counterexamples in positive characteristic by producing compact $A$-orbits on which $\mu$ attains the conjectured upper bound, so the conjecture is refuted.