Nienhuis's BKT conjecture for O(n) models

Consider the $O(n)$ model on the hexagonal lattice with loop-weight $n\in(0,2)$ and edge-weight $x>0$, and define

$$x_c(n)=\frac{1}{\sqrt{2+\sqrt{2-n}}}.$$

Nienhuis's BKT conjecture. For $x<x_c(n)$, the probability that two points lie on the same loop decays exponentially fast, whereas for $x\geq x_c(n)$ it decays as a power law. The source says this has been rigorously established only for $n=0$ and $n=1$; the remaining cases are open.