Period equality conjecture for the hunger game

Let a finite irreducible Markov chain have rational transition probabilities, stationary distribution $\pi$, and let $n$ be the least common denominator of the entries of $\pi$. A hunger-game state is recurrent if it returns to itself under the hunger game, and the period of a cycle is its number of steps. Period equality conjecture. Every cycle in the hunger game for the chain has period $n$. This conjecture generalizes the proved fact that the zero hunger vector has period $n$; the source presents the equality of all cycle periods as an empirical observation, and no resolution is given.