Batchelor–Yung surface fugacity conjecture for the honeycomb O(n) loop model

For the $O(n)$ loop model on the semi-infinite hexagonal lattice with $n\in[-2,2]$, associate a fugacity $x_{\rm c}(n) = 1/\sqrt{2+\sqrt{2-n}}$ with occupied vertices and an additional fugacity $y$ with occupied vertices on the boundary. Batchelor–Yung conjecture. The model undergoes a special surface transition at

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

Batchelor and Yung obtained this value using integrability and assumptions analogous to those used by Nienhuis for the bulk critical point. The conjecture concerns the adsorption transition of the honeycomb-lattice $O(n)$ model for $n\in[-2,2]$.