  We consider the following question by Balister, Gy\H{o}ri and Schelp: given
$2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always
possible to partition the elements of $\mathbb{F}_2^n$ into pairs such that the
difference between the two elements of the $i$-th pair is equal to the $i$-th
given vector for every $i$? An analogous question in $\mathbb{F}_p$, which is a
case of the so-called "seating couples" problem, has been resolved by
Preissmann and Mischler in 2009. In this paper, we prove the conjecture in
$\mathbb{F}_2^n$ in the case when the number of distinct values among the given
difference vectors is at most $n-2\log n-1$, and also in the case when at least
a fraction $\frac12+\varepsilon$ of the given vectors are equal (for all
$\varepsilon>0$ and $n$ sufficiently large based on $\varepsilon$).
