A note on optimization of the second positive Neumann eigenvalue for parallelograms

It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to twice the other. In this note we prove that this conjecture is true within the class of parallelogram domains.