Nienhuis's critical-value conjecture for the O(n) loop model

Let $n\in[0,2]$ be the loop weight and $x>0$ the edge weight in the hexagonal-lattice $O(n)$ loop model, whose configuration weights are proportional to $n^{\mathrm{\#~loops}}x^{\mathrm{\ length~of~loops}}$. Nienhuis's conjecture. The critical value is

$$x_{c}(n)=\frac1{\sqrt{2+\sqrt{2-n}}}~.$$

This conjecture identifies the transition between the dilute and dense phases; the criticality was rigorously established only for $n=1$ in the stated discussion.