Characterization of critical positions in game NIM$(n,k)$

Let NIM$(n,k)$ be the impartial game with positions $x=(x_1,\dots,x_n)$ and moves reducing between one and $k$ coordinates, and let an $m$-critical position mean a position whose remoteness level is $m$. For integers $k,n,m$ satisfying $0<k<n$ and $m\geq 0$, every $m$-critical position satisfies

$$km\leq x_1+\dots+x_n<k(m+1)$$

and

$$\max(x_1,\dots,x_n)\leq m.$$

Characterization conjecture. The displayed inequalities characterize the $m$-critical positions of NIM$(n,k)$.

The paper notes that characterizing $m$-critical positions of NIM$(n,k)$ remains open; the displayed conditions are presented as a conjectural generalization of the established results for the case $n=k+1$.