Tóth's infinite-cycle conjecture for the interchange process

Let $\pi _\beta$ be the interchange permutation on $\mathbb Z^d$ at time $\beta>0$, and let $\beta_c$ denote the critical time.

Tóth's conjecture. Almost surely, $\pi_\beta$ has only finite cycles when $d=2$, for every $\beta>0$, whereas when $d\geq3$ it has infinite cycles for every $\beta>\beta_c$.

The paper proves the infinite-cycle assertion for $d\geq5$ and all sufficiently large $\beta$, but the conjecture remains open in the stated generality, particularly in dimensions three and four.