Balister–Győri–Schelp partition conjecture for binary vector lists

Let $k\geq 2$, and let $v_1,\dots,v_{2^{k-1}}$ be nonzero vectors in $\mathbb F_2^k$ satisfying

$$\sum_{i=1}^{2^{k-1}}v_i=0.$$

A 2-set of vectors is a pair of distinct vectors, and a partition of $\mathbb F_2^k$ into such 2-sets consists of pairs $\{w_i,z_i\}$ whose union is $\mathbb F_2^k$.

Balister–Győri–Schelp conjecture. There exists a partition of $\mathbb F_2^k$ into 2-sets $\{w_i,z_i\}$, for $i\in\{1,\dots,2^{k-1}\}$, such that

$$v_i=w_i+z_i$$

for all $i$.

The conjecture is connected with matching and rainbow-matching problems in finite abelian groups. The source reports partial results, including an asymptotic bound for the corresponding functional batch code, but does not state a complete resolution.