On a conjectured reverse Faber-Krahn inequality for a Steklov-type Laplacian eigenvalue

For a given bounded Lipschitz set Ω, we consider a Steklov-type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding H^1(Ω)↪L^2(∂Ω). We prove that a conjectured reverse Faber-Krahn inequality holds true at least in the class of Lipschitz sets which are "close" to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.