Kipnis–Varadhan quenched functional CLT conjecture

Let $(X_n)_{n\in\mathbb{Z}}$ be a stationary reversible Markov chain with transition operator $Q$, invariant distribution $\pi$, and let $f\in\lbrack(I-Q)\mathbb{L}_{q}(\pi)]\cap\mathbb{L}_{p}^{0}(\pi)$ for $p\in\lbrack2,\infty)$ and $1/p+1/q=1$. Kipnis–Varadhan conjecture. If

$$\mathbb{E}_{\pi}\mathbb{(}f\sum\nolimits_{j=0}^{n}Q^{j}f)\text{ is convergent.}$$

Then the quenched functional CLT holds. The conjecture asks whether the functional central limit theorem of Kipnis and Varadhan holds in the quenched sense for reversible Markov chains; the source states that it remains unsolved.