Square minimizes the lowest magnetic Dirichlet eigenvalue among rectangles with fixed perimeter

Establish that, among rectangles with fixed perimeter in the plane and for any constant magnetic field B ∈ ℝ, the square minimizes the lowest eigenvalue of the magnetic Dirichlet Laplacian (−i∇−A)² with Dirichlet boundary conditions.

Background

Beyond the fixed-area formulation, the authors explicitly state a perimeter-constrained version of the optimization: the square should minimize the principal magnetic Dirichlet eigenvalue among rectangles with fixed perimeter. They note this perimeter version is implied by the area version via scaling and the classical geometric isoperimetric inequality, but since the area conjecture is unresolved in general, the perimeter variant remains conjectural as well.