Random-current phase transition conjecture on the hexagonal lattice

Let $\mathbb{H}$ be the hexagonal lattice. Denote by $\mathbf{P}_{\mathbb{H}}$ the single random-current measure and by $\phi_{\mathbb{H}}$ the associated random-cluster model.

Hexagonal-lattice random-current conjecture. All phase transitions of the random current and random-cluster model on the hexagonal lattice coincide:

$$\beta_c^{\operatorname{perc}}(\mathbf{P}_{\mathbb{H}})=\beta_c(\phi_{\mathbb{H}}).$$

The loop $\mathrm{O}(1)$ model has an explicitly understood phase diagram on the hexagonal lattice, whereas the random-current measure is expected to exhibit a more generic behaviour. The conjectured equality remains open in the source.