Uniqueness conjecture for bounded symmetric domains with the same harmonic measure distribution

Let $\Omega$ and $\widetilde\Omega$ be domains that are both bounded and symmetric in the sense of the paper's definition. Let $h_\Omega$ and $h_{\widetilde\Omega}$ denote their harmonic measure distribution functions.

Uniqueness conjecture. If

$$h_\Omega=h_{\widetilde\Omega},$$

then $\Omega=\widetilde\Omega$ up to a set of harmonic capacity zero.

This conjecture proposes that boundedness and symmetry eliminate the non-uniqueness caused by rotations, reflections, and deletion of sets of harmonic capacity zero. The surrounding text establishes uniqueness for circle domains and motivates the conjecture, but provides no resolution in the stated generality.