Fine torus conjecture for discretizing integral seminorms

Let $K\subseteq\mathbb R^d$ be a nondegenerate integral convex polytope containing the origin. A fine structure on $\mathbb R^d$ is required to be $\mathbb Z^d$-periodic, and $d_{\widetilde F}$ denotes its induced metric. Fine torus conjecture. There exists such a fine structure $\widetilde F$ satisfying

$$d_{\widetilde F}(x,x+v)=\max_{\varphi\in K}\varphi(v)=:\|v\|_K$$

for every $v\in\mathbb Z^d$ and every vertex $x$ of $\widetilde F$, with exactly $d!^2|K|$ $d$-dimensional cells up to integer translations. The source presents this as a conjectural optimal discretization of integral seminorms; the discretization problem is noted to be open already in dimension three for the cube, where the conjectured minimum is 36 tetrahedra per period.