Has there been any progress on the 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].

Edit 1: 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