Wilson's conjecture on critical XOR-Ising loops

Let the critical XOR-Ising model be defined on the honeycomb lattice, and consider its family of loops. The Gaussian free field is normalized in the usual way, and a level line is a contour corresponding to a specified field level.

Wilson's conjecture. The scaling limit of the family of loops of the critical XOR Ising model are the level lines of the Gaussian free field corresponding to levels that are odd multiples of $\frac{\sqrt{\pi}}{2}$.

Wilson's conjecture is motivated by numerical simulations and by the convergence of the associated dimer-model height function to $\frac{1}{\sqrt{\pi}}$ times the Gaussian free field. The paper describes this as a conjecture; convergence of the height function alone is too weak to establish convergence of contour lines.