P-position semilinearity conjecture for Exact Slow $k$-Nim

Let ${\sc Nim}^1_{n,=k}$ be the impartial game on nonnegative integer vectors in which a move decreases exactly $k$ coordinates by one. A position from which the previous player can force a win is called a P-position. A set of integer vectors is semilinear if it can be expressed in Presburger arithmetic, equivalently as a finite union of solution sets of systems of linear inequalities and congruences. Semilinearity conjecture. For any $n$ and $k$, the P-positions of ${\sc Nim}^1_{n,=k}$ form a semilinear set. The explicit characterization for ${\sc Nim}^1_{6,=2}$ gives a polynomial-time decision procedure, motivating the conjecture for all values of $n$ and $k$.