Balister–Győri–Schelp conjecture on Hadamard pair solutions

Throughout, all vectors and matrices are over $\mathbb{F}_2$. Fix $s\geqslant 2$ and let $M=[v_0,\ldots,v_{2^{s-1}-1}]$ be an $s\times 2^{s-1}$ matrix with nonzero columns. Let $\mathcal{H}=[h_0,\ldots,h_{2^s-1}]$ be an $s\times 2^s$ matrix whose columns are precisely the vectors of $\mathbb{F}_2^s$, each appearing once. A Hadamard pair solution for $M$ is an ordering of the columns of $\mathcal{H}$ such that $v_i=h_{2i}+h_{2i+1}$ for every $i\in\{0,\ldots,2^{s-1}-1\}$.

Balister–Győri–Schelp conjecture. If

$$\sum_{i=0}^{2^{s-1}-1}v_i=0,$$

then $M$ has a Hadamard pair solution.

The zero-sum condition is necessary because the sum of all vectors in $\mathbb{F}_2^s$ is zero. The conjecture asserts that this necessary condition is also sufficient for partitioning $\mathbb{F}_2^s$ into pairs with the prescribed nonzero differences; the supplied source does not state whether it has been resolved.