Tóth's phase-transition conjecture for spontaneous magnetization

Let $m(\beta)$ denote the spontaneous magnetization of the quantum Heisenberg ferromagnet, represented through the interchange process on $[-r,r]^3$ at inverse temperature $\beta$. Let $c_\beta(0)$ be the size of the cycle containing $0$ at time $\beta$, and let $s_k(\beta)$ be the number of cycles of length $k$. Tóth's conjecture. The function $m(\beta)$ admits a phase transition: there exists some $\beta_c$ such that

$$m(\beta)=0\quad\text{for }\beta<\beta_c,$$

and

$$m(\beta)>0\quad\text{for }\beta>\beta_c.$$

This conjecture concerns the existence of an ordered, or high-$\beta$, phase in the quantum Heisenberg ferromagnet. The existence of that phase remains open; the main unresolved issue is not sharpness or uniqueness of the transition, but whether the high-$\beta$ phase exists at all.