Ergodicity conjecture for convergence of the information state

Consider a positive, anchored hidden Markov model, and let $F|_R$ be the restricted information chain on the recurrent state set $R$. Theorem 2 states that irreducibility and aperiodicity of $F|_R$ suffice for convergence in distribution of the information state $Z_t$ to a discrete invariant measure $\mu_\infty\in\mathcal P(R)\subset\mathcal P(\mathcal P(S))$.

Ergodicity conjecture. The conditions of that theorem can be weakened to the case when the underlying chain is only ergodic.

If true, this would extend the convergence result beyond the stated irreducibility and aperiodicity assumptions on the restricted information chain. The excerpt does not establish the weakening.