Equivalence of finiteness of expected attraction time across initial states

Establish that for any Markov random dynamical system on a countable state space representing an irreducible and positive recurrent Markov chain, the expected hitting time E[T_A(ω, x)] of the random attractor A is finite for one initial state x in X if and only if it is finite for all initial states in X.

Background

The paper proves that the expected time to hit the random attractor A is finite for all initial states if and only if the associated Markov chain is ergodic of degree 2. However, the authors cannot rule out the possibility that, for chains not ergodic of degree 2, some initial states have finite expected hitting time while others do not.

Motivated by this gap, they formulate a conjecture asserting an all-or-nothing property: finiteness of E[T_A(ω, x)] for a single initial condition should already imply finiteness for every initial condition, and conversely.