Nash equilibrium conjecture for the GM-strategies in exact slow NIM

Let $x$ be a position in the exact slow game $\operatorname{NIM}(n,n-1)$ played by $\ell$ players in a fixed cyclic order. A move reduces $n-1$ positive entries by $1$ and leaves one entry unchanged; terminal positions have at least two non-positive entries. The GM-rule assigns a strategy to each player, including the rule that when no entry of $x$ is a multiple of $\ell$, the largest entry is kept unchanged and the other $n-1$ entries are reduced by $1$. Nash equilibrium conjecture. The set of $\ell$ GM-strategies form a uniform Nash equilibrium. The claim concerns the strategic behavior of the exact slow NIM game with multiple players and asserts that the GM-strategies are jointly stable under unilateral deviation. The supplied text does not state whether this has been proved or remains open.