Critical surface fugacity conjecture for the rotated honeycomb $O(n)$ loop model

Let the $O(n)$ loop model be defined on the semi-infinite honeycomb lattice with the boundary oriented as in Figure~(b). For $n\in[-2,2]$, assign fugacity

$$x_{\rm c}(n)=\frac{1}{\sqrt{2+\sqrt{2-n}}}$$

to occupied vertices and an additional fugacity $y$ to occupied boundary vertices. Critical surface fugacity conjecture. The model undergoes a surface transition at

$$y=y_{\rm c}(n)=\sqrt{\frac{2+\sqrt{2-n}}{1+\sqrt{2-n}-\sqrt{2+\sqrt{2-n}}}}.$$

The value is proved to be the critical surface fugacity for self-avoiding walks when $n=0$; the conjecture extends this result to the full range $n\in[-2,2]$ for the $O(n)$ loop model.