General-domain hole probability conjecture for the Gaussian entire function

Let $f$ be the Gaussian entire function, let $K\subset\mathbb{C}$ be a connected compact set with non-empty interior, and let $rK$ denote the homothety of $K$ by $r$. Write

$$p_{H,K}(f;r)=\log^{-}\mathcal{P}\{f(z)\ne0\text{ in }rK\},$$

and let $S(r)$ be the hole-probability scale for the Gaussian entire function. General-domain hole probability conjecture. For large values of $r$,

$$p_{H,K}(f;r)=S\left(c(K)\cdot r\right)\left(1+o\left(1\right)\right),$$

where $c(K)$ is the capacity (transfinite diameter) of $K$. The conjecture extends the known formulation for standard domains to arbitrary connected compact domains; its motivation comes from the plane-isometry invariance of the zero set and the appearance of the Vandermonde determinant in the relevant estimates. The supplied text does not state a resolution, so the conjecture remains open.