Percolation and circle-packing modulus conjecture for periodic triangulations

Let $T^*$ be a triangulation of the torus, and let $\hat T^*$ be its universal cover. For $\alpha\in\mathbb{H}$, write $\hat T^*_{\alpha}$ for the corresponding realization in the conformal modulus $\alpha$. Percolation modulus conjecture. The critical parameter for site-percolation on $\hat T^*$ is $1/2$, and for every $\alpha\in\mathbb{H}$, critical site-percolation on $\hat T^*_{\alpha}$ has a scaling limit. Moreover, the modulus at which the model is conformally invariant in the scaling limit is the modulus obtained from the circle packing associated to $\hat T^*$:

$$\alpha_T^{\mathrm{Perc}}=\alpha_T^{\mathrm{CP}}.$$

This conjecture proposes a universality principle for critical site-percolation on periodic triangulations and identifies its conformally invariant embedding with the circle-packing embedding. The source explicitly describes it as closer to wishful thinking, and gives no resolution.