Courant-sharpness conjecture for negative Robin eigenvalues of the square

Let $S$ be the square, let $q$ be an even integer with $q\geq 4$, and let $\lambda_{0,q,h}(S)$ denote the corresponding Robin eigenvalue for parameter $h$. An associated eigenfunction is an eigenfunction belonging to this eigenvalue.

Courant-sharpness conjecture. If $\lambda_{0,q,h}(S)$ is negative, then no associated eigenfunction is Courant-sharp.

This conjecture concerns the possible Courant-sharp Robin eigenvalues of the square in the negative-parameter regime. The supplied text presents it as the main conjecture after proving a related proposition for the case $q=4$; its general status is not specified here.