Small-parameter stability conjecture for Courant-sharp Robin eigenvalues on the square

Let the Robin eigenvalues on the square be indexed in increasing order, with Robin parameter $h>0$, and call an eigenvalue Courant-sharp when it has an eigenfunction whose number of nodal domains equals its index. The case $h=0$ is the Neumann problem.

Small-parameter stability conjecture. There exists $h_0>0$ such that for $0<h\leq h_0$, the Courant-sharp cases for the Robin problem are the same, except the fifth one, as those for $h=0$ (i.e. the Neumann case).

This conjecture concerns the behavior of Courant-sharp Robin eigenvalues as the Robin parameter tends to zero. It is proposed for future work; the supplied text gives no resolution.