Cyclicity conjecture for the periodicity group

Let $\eqp$ be an irreducible ergodic quantum process, and let $\GammaGroup$ denote its periodicity group. Cyclicity conjecture. The group $\GammaGroup$ is cyclic.

The group is finite of order at most $d^2$; proving cyclicity would imply that its order is at most $d$. The supplied text identifies this as an unsolved conjecture.