The critical threshold conjecture for Bargmann-Fock percolation

Let $\mathcal{D}_p$ denote the excursion domain at level $p$ associated with a centered, normalized, non-degenerate, sufficiently smooth, stationary, isotropic, and positively correlated random field on $\mathbb{R}^2$ with sufficient correlation decay. Critical threshold conjecture. The probability that $\mathcal{D}_p$ has an unbounded connected component is $1$ if $p>0$, and $0$ otherwise. This conjecture predicts that the self-dual level $p=0$ is the critical threshold for planar Gaussian percolation, analogously to the critical parameter $p=1/2$ in planar Bernoulli percolation. The cited context does not indicate that the conjecture has been resolved.