Bogomolny–Schmit conjecture for random-plane-wave nodal geometry

Let the random plane wave (RPW) be the stationary Gaussian field in the plane discussed in the paper, and consider its nodal lines and nodal domains at level zero. Let critical percolation denote critical planar percolation, and let $\operatorname{CLE}(6)$ denote the conformal loop ensemble with parameter $6$. Bogomolny–Schmit conjecture. All large-scale connectivity properties of RPW nodal lines and domains are the same as for critical percolation. In particular, all crossing events have the same scaling limits, and the collection of nodal lines has a conformally invariant scaling limit given by $\operatorname{CLE}(6)$. This conjecture is explicitly described as wide open, with only numerical evidence and heuristics in the source.