Is Markov reachability problem Skolem-Hard for Ergodic Markov chains?

Given a Markov chain $M$, an initial distribution $u$ and target distribution $v$, and a rational number $r$, consider the problem of checking if there exists an $n$ such that $u M^n v =r$. This problem is known to be as hard as the Skolem problem of LRS. However, the reduction requires the Markov chain to be non-ergodic, in particular it is reducible and periodic. If we now assume the Markov chain to be irreducible and aperiodic (a natural assumption that is often used since forever!), does it remain hard?! Both yes or no answers would be very interesting and have consequences to model checking of probabilistic linear dynamical systems.