The squared coefficient-of-variation conjecture for irreducible MMPPs

An MMPP is a Markovian modulated Poisson process; write $c^2$ for the squared coefficient of variation of its interarrival time. Squared coefficient-of-variation conjecture. For an irreducible MMPP, $c^2\geq 1$. The claim is motivated by a folklore assumption and is known for the two-state case, but no proof is known for an arbitrary irreducible MMPP.