I am reposting a previous question due to incorrect initial formulation.

Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = \lambda^t f$. Let the chain start at a single state $x \in \Omega$. Is it true that $\mathbb{E}_x (P_tf- \mathbb{E}_x( P_t f))^2$ is nondecreasing in t? As pointed out by James Martin, it's important to start with the point mass distribution at a single state, since otherwise one could choose an initial distribution that has an f-variance larger than the stationary $f$-variance.

Wilson's method gives a way to bound the variance of eigenfunctions in t in a way that's reminiscent of the Martingale difference method, but since the variance at time $\infty$ is usually easy to calculate, if we know the above monotonicity result, we could bound the variance at finite time easily.