Day–Johnson nonexistence conjecture for equilibria in hypercube Voronoi games

Let $Q_n$ be the discrete $n$-dimensional hypercube, and let a $k$-player profile be a tuple of $k$ chosen vertices of $Q_n$. A profile is in equilibrium if no player can strictly increase their score by moving to a different vertex. Day–Johnson's conjecture. For any fixed $k\geq 5$, there exists some $N$ such that, for any $n\geq N$, there are no $k$-player profiles in equilibrium on $Q_n$. The conjecture asserts eventual nonexistence of equilibria for every fixed number of at least five players; it remains open for all $k\geq 5$.