Eulerian percolation threshold conjecture on the square lattice

Let $p_{c,\mathrm{even}}$ denote the percolation threshold for Eulerian percolation on $\mathbb{Z}^2$, let $\mu_p$ be the corresponding Eulerian percolation measure, let $\mathcal{C}$ be the event that an infinite cluster exists, and let $\beta_c$ be the critical inverse temperature of the Ising model. Eulerian percolation threshold conjecture.

$$\mu_{1-p_{c,\mathrm{even}}}(\mathcal{C})=1.$$

Equivalently, for every Gibbs measure with parameter $-\beta_c$, contours almost surely percolate.

The conjecture concerns the unresolved endpoint $p=1-p_{c,\mathrm{even}}$, corresponding to $\beta=-\beta_c$ in the Ising model. Conditioning on the Eulerian constraint destroys the usual monotonicity arguments available for independent Bernoulli and FK percolation.