Sharp percolative transition conjecture for the single random current on $\mathbb{Z}^d$

Let $d\geq 3$, and let $\mathbf{P}$ denote the single random-current measure on $\mathbb{Z}^d$. Let $\beta_c^{\operatorname{perc}}(\mathbf{P})$ be its percolation threshold and $\beta_c^{\exp}(\mathbf{P})$ its exponential-decay threshold.

Random-current sharp-transition conjecture. The single random current on $\mathbb{Z}^d$ has a unique sharp percolative phase transition, namely

$$\beta_c^{\operatorname{perc}}(\mathbf{P})=\beta_c^{\exp}(\mathbf{P}).$$

This is presented as a weaker alternative to the corresponding conjecture for the loop $\mathrm{O}(1)$ model. It remains open in dimensions at least three.