Conjecture on Steklov spectral determination of convex polygonal domains

Let $P$ be a convex polygonal domain, and let $\Omega$ be a simply-connected smoothly bounded domain. Two domains are Steklov isospectral if their Steklov spectra, counted with multiplicities, coincide. Steklov spectral determination conjecture. A convex polygonal domain cannot be Steklov isospectral to a simply-connected smoothly bounded domain. This conjecture concerns whether the Steklov spectrum can distinguish convex polygonal domains from simply-connected smoothly bounded domains; the source presents it as a conjecture based on the evidence developed in the section, with its general resolution left open.