Has there been any progress on the $\textit{smooth}$ isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit disk?

Related to [this][1] and of course [this][2].

$\textit{Edit1:}$ What if a non-local boundary condition is enforced? To be precise, suppose $D$ is an open disk, and $\Omega$ is a symmetric bounded domain with smooth boundary so that $D$ and $\Omega$ are isospectral w.r.t to Laplacian with non-local boundary condition. Does it follow that $D$ and $\Omega$ are congruent? [1]: Can one hear the shape of a drum for operators? [2]: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarkKac.pdf