Tóth's phase-transition conjecture for long cycles in the three-dimensional interchange process

Let $\beta>0$ be the inverse-temperature parameter, and let $\sigma_\beta$ denote the permutation generated by the interchange process on the lattice $\{-n,\dots,-1,1,\dots,n\}^3$. For a fixed $c\in(0,1)$, consider the limiting probability that the largest cycle of $\sigma_\beta$ has length greater than $c(2n)^3$.

Tóth's conjecture. The quantity

$$\lim_{n\rightarrow\infty}\mathbb{P}\bigl(\text{the length of the largest cycle of }\sigma_\beta>c(2n)^3\bigr)$$

undergoes a phase transition as a function of $\beta$: it is zero for $\beta<\beta'$ and positive for $\beta>\beta'$, where $\beta'$ is the critical inverse temperature.

This conjecture is motivated by Tóth's representation formula for the spontaneous magnetization of the quantum Heisenberg ferromagnet and proposes an analogous phase transition for long cycles in the three-dimensional interchange process. The supplied source does not indicate whether the conjecture has been resolved.