Generalized Pansu conjecture for sub-Finsler Heisenberg groups

Let $(H(\mathbb R),d_Q,\lambda)$ be the Heisenberg group equipped with Haar measure $\lambda$ and an arbitrary Carnot–Carathéodory metric $d_Q$ arising from a norm on the horizontal plane. Surface area is taken to be either Minkowski content or anti-Minkowski content associated with the dual metric. Generalized Pansu conjecture. For any such $d_Q$ and either notion of surface area, the Pansu bubble set is the unique isoperimetrix, up to dilation and translation. The conjecture extends Pansu's proposed characterization from the sub-Riemannian setting to arbitrary sub-Finsler Carnot–Carathéodory metrics. The source presents computations suggesting the Pansu bubble set is superior to several generalized bubble sets, but does not establish the conjecture.