Translation-tiling conjecture for hunger-game basins of attraction

Let a finite rational irreducible Markov chain have $n$ states, transition matrix with hunger-game matrix $H$, and basin of attraction in the hyperplane $Z$. The vectors $H_1-H_2,H_2-H_3,\dots,H_{n-1}-H_n$ are regarded as vectors in $Z\subseteq\mathbb{R}^n$. Translation-tiling conjecture. The basin of attraction tiles the hyperplane $Z$ by translation, and the translation vectors relating the tiles form an $(n-1)$-dimensional sublattice of $Z$ with basis vectors

$$H_1-H_2,H_2-H_3,\dots,H_{n-1}-H_n.$$

The conjecture formalizes the observed tiling in an example and extends it to general finite rational irreducible Markov chains; the source gives no resolution.