Loop O(n) phase transition for all loop weights

Determine the critical curve $x_c(n)$ governing the loop-length phase transition of the loop $O(n)$ model for every $n\in(0,2]$, including the transition between macroscopic loops for $x\geq x_c(n)$ and exponential loop-length decay for $x<x_c(n)$.

Background

The loop $O(n)$ model is defined on the hexagonal lattice with loop-weight parameter $n>0$ and edge-weight parameter $x>0$. The paper explains that the model lacks monotonicity and is conjectured to undergo a loop-length phase transition throughout the regime $n\in(0,2]$. The conjectural critical curve is later identified as $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$, while the paper proves macroscopic behavior only in a substantial subregion of the predicted phase diagram.