Uniform 2-partition conjecture for the second mixed eigenvalue of the disk

Let the unit disk carry mixed Dirichlet--Neumann boundary conditions, with the Dirichlet part varying among boundary arrangements; the uniform $2$--partition is the arrangement consisting of two equally sized, uniformly distributed Dirichlet parts. Uniform 2-partition conjecture. The minimizing arrangement of boundary conditions for the second eigenvalue of the mixed Dirichlet--Neumann problem on the disk is given by the uniform $2$--partition. The conjecture concerns the minimization of the second mixed eigenvalue under variation of the boundary arrangement; the surrounding discussion says that it will be justified in the paper, but the supplied text does not state whether a proof is completed.