The exclusion-voter passage-time moment conjecture

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$, state space $\mathcal{D}$, distinguished state $\mathcal{D}_0$, and relaxation time

$$\tau:=\min\{t\in\mathbb{N}:\xi_t=\mathcal{D}_0\}.$$

The passage-time moment conjecture. Suppose that $p\leq1/2$ and $\beta\in[0,1]$. For any $\varepsilon>0$ and $S_0\in\mathcal{D}\setminus\{\mathcal{D}_0\}$,

$${\mathbb{E}}_{\beta,p}\bigl[\tau^{(3/2)+\varepsilon}\mid\xi_0=S_0\bigr]=\infty.$$

The conjecture asserts that mixing transient exclusion with the voter model does not produce a lighter passage-time tail. The paper describes it as challenging because exclusion moves can increase the number of blocks; the stated $2+\varepsilon$ moment divergence is proved, but the exponent $3/2$ remains open.