Absence of unbounded nodal components at criticality for planar Gaussian fields

Let $f$ be a Gaussian field satisfying Assumption $\mathrm{}$ that is $D_4$-symmetric (e.g. $f$ is the RPW). The nodal set $\{f=0\}$ is the level set at criticality.

Critical nodal-set conjecture. The set $\{f=0\}$ has no unbounded connected components.

This is the fundamental question of whether nodal lines are bounded, equivalently whether $\{f\geq 0\}$ is bounded. The claim is known when $\kappa\geq 0$ and in a perturbative non-FKG regime, but remains open for important examples such as the random plane wave.