  We consider the Laplace operator in the exterior of a compact set in the
plane, subject to Robin boundary conditions. If the boundary coupling is
sufficiently negative, there are at least two discrete eigenvalues below the
essential spectrum. We state a general conjecture that the second eigenvalue is
maximised by the exterior of a disk under isochoric or isoperimetric
constraints. We prove an isoelastic version of the conjecture for the exterior
of convex domains. Finally, we establish a monotonicity result for the second
eigenvalue under the condition that the compact set is strictly star-shaped and
centrally symmetric.
