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

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


  [1]: http://mathoverflow.net/questions/223672/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