The GM-rule Nash equilibrium conjecture for exact slow NIM

Consider the exact slow game NIM$(n,n-1)$ for $\ell$ players: a position is a non-negative $n$-vector, players move cyclically, and on each move a player keeps one entry unchanged and decreases each of the other $n-1$ positive entries by $1$. A position with at least two non-positive entries is terminal; the player whose turn it is loses, and the other $\ell-1$ players share the payoff $\frac{C-L(P)}{\ell-1}$, where $L(P)$ is the play length and $C$ exceeds the length of every play from the initial position. The GM-rule specifies a strategy for each player, including the move that keeps the largest entry when no entry is a multiple of $\ell$. GM-rule Nash equilibrium conjecture. The set of $\ell$ GM-strategies forms a uniform Nash equilibrium. The claim concerns the strategic behavior induced by the GM-rule in the multiplayer exact slow NIM game; the supplied text does not state whether it has been proved or disproved.