Holroyd's interval hitting-time conjecture for finite Markov chains

Let $X$ be a Markov chain with finite state space $S$, where $|S|=n$, and let $x$ and $y$ be any two states. Holroyd's interval hitting-time conjecture. For all $t>n$,

$$\mathbf{P}_x(t\leq\tau(y)\leq t+n)=O\left(\frac{n}{t}\right).$$

This conjecture asks for an interval bound on hitting-time probabilities without the periodicity present in the paper's extremal example. It is attributed to Holroyd and is presented as an open question; no resolution is given in the source.