Percolation conjecture for a single current above the Ising critical point

Let $\beta_c$ be the critical inverse temperature, and let ${\bf P}_{\mathbb Z^2,\beta}^{\emptyset}$ denote the infinite-volume sourceless random-current measure on $\mathbb Z^2$. Write $0\xleftrightarrow{}\infty\text{ in }\widehat{\mathbf n}$ for the event that the origin is connected to infinity in the current configuration $\widehat{\mathbf n}$. Single-current percolation conjecture. For $\beta>\beta_c$, one has

$${\bf P}_{\mathbb Z^2,\beta}^{\emptyset}[0\xleftrightarrow{}\infty\text{ in }\widehat{\mathbf n}]>0.$$

The conjecture would strengthen the known supercritical behavior for the sum of two independent currents: $\beta_c$ is the phase transition for percolation of $\mathbf n_1+\mathbf n_2$, whereas the corresponding statement for a single current remains open. It is motivated by the possibility that above $\beta_c$, current loops unfold into infinite loops.