Modified Erdős-Ginzburg-Ziv constants for ${\mathbb{Z}}_2^d$

Let $G$ be a finite abelian group written additively, and let $r$ be a multiple of its exponent. The modified Erdős-Ginzburg-Ziv constant ${\mathsf{s}}_r^{\mathrm{\prime }}(G)$ is the smallest integer $s$ such that every zero-sum sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We find exact values of ${\mathsf{s}}_{2k}^{\mathrm{\prime }}({\mathbb{Z}}_2^d)$ for $d\le 2k+1$.