Reversible Markov-chain surprise bound

Let $X$ be a reversible Markov chain with finite state space, and let $\pi$ be the stationary distribution of $X$. For states $x$ and $y$, write $\tau(y)$ for the hitting time of $y$ starting from $x$. Reversible surprise conjecture. For all $t>0$,

$$\mathbf{P}_x(\tau(y)=t)=O\left(\frac{1}{t}\sqrt{\max\left(1,\log\frac{1}{\pi(x)}\right)}\right).$$

This conjecture would improve the bound in the paper's theorem for reversible chains and is motivated by an example showing a gap between the known upper bound and the currently available lower-bound construction. Its resolution is not supplied here.