Two disks maximize the third Robin eigenvalue: positive parameters

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in $[-4\pi ,4\pi ]$. This sharp inequality was known previously only for negative parameters in $[-4\pi ,0]$, by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in $(0,4\pi ]$ by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.