The disk's nonexistence conjecture for Dirichlet-Neumann isospectrality

A bounded planar domain is equipped with a decomposition of its boundary into Dirichlet and Neumann parts, and Dirichlet-Neumann isospectrality means that swapping these boundary conditions leaves the spectrum unchanged. The disk is the planar domain under consideration. The disk's nonexistence conjecture. A disk does not admit Dirichlet-Neumann isospectrality. This asks whether there are domains with no non-trivial Dirichlet-Neumann isospectral decompositions; the source presents the disk as a natural candidate, but gives no resolution.