The Davenport constant of a box

Given an additively written abelian group $G$ and a set $X\subseteq G$, we let $\mathscr{B}(X)$ denote the monoid of zero-sum sequences over $X$ and $\mathsf{D}(X)$ the Davenport constant of $\mathscr{B}(X)$, namely the supremum of the positive integers $n$ for which there exists a sequence $x_1 \cdots x_n$ of $\mathscr{B}(X)$ such that $\sum_{i \in I} x_i \ne 0$ for each non-empty proper subset $I$ of $\{1, \ldots, n\}$. In this paper, we mainly investigate the case when $G$ is a power of $\mathbb{Z}$ and $X$ is a box (i.e., a product of intervals of $G$). Some mixed sets (e.g., the product of a group by a box) are studied too, and some inverse results are obtained.