Semilinearity conjecture for nonnegative FDG games

Let $d$ be a positive integer and let ${\cal D}\subset\mathbb{Z}^{d}$ be a finite difference set satisfying

$$\sum_{i=1}^d a_i>0$$

for every $(a_1,\dots,a_d)\in{\cal D}$. The positions are vectors $x\in\mathbb{N}^{d}$, and a move from $x$ to $y$ is allowed when $x-y\in{\cal D}$. Such a game is called an FDG game. Nonnegative FDG semilinearity conjecture. For any FDG game such that $a_i\geq0$ for each $(a_1,\dots,a_d)\in{\cal D}$, the set of P-positions is semilinear. This is proposed as a stronger version of the Exact Slow $k$-Nim conjecture because ${\sc Nim}^1_{n,=k}$ has difference vectors with nonnegative coordinates. The contrast with general FDG games, where nonsemilinearity and computational hardness can occur, highlights the role of the nonnegativity assumption.