Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ is aperiodic, irreducible and additionally positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true that for any $x\in S$ $$\underset{n\rightarrow\infty}{\text{lim}}~\underset{\text{A}\in\mathcal{F}^{\mathbb{Z}_+}}{\text{sup}}~|P_x((X_{n},X_{n+1},\dots)\in\text{A})-P_\pi(X\in A)| =0?$$
I would assume this is true since as a positive recurrent chain it keeps hitting again and again also in its asymptotics. But I can't figure out how to prove it. I tried to come up with any clever coupling which seems to be the way to deal with those Variance norms (see Durrett Chapter 6, Convergence Theorem).