The asymptotic Davenport constant conjecture for Euclidean balls

Let $d\geq 2$ be an integer, let $\mathcal{B}^{(d)}_m$ denote the integer points in the $d$-dimensional Euclidean ball of radius $m$, and let $\mathsf{D}^{(d+1)}(\mathcal{B}^{(d)}_m)$ be the maximal length of a minimal zero-sum sequence over this set with support of size at most $d+1$.

Asymptotic ball Davenport conjecture. For $m\geq\sqrt d$,

$$\mathsf{D}(\mathcal{B}^{(d)}_m)=\mathsf{D}^{(d+1)}(\mathcal{B}^{(d)}_m)\underset{m\to+\infty}{\sim}\left(\frac{(d+1)^{d+1}}{d^d}\right)^{1/2}m^d.$$

The known two- and three-dimensional results motivate the assertion that extremal minimal zero-sum sequences in every dimension have support of size at most $d+1$ and that their asymptotic size is governed by a regular simplex. The general statement remains open.