Pansu's bubble-set conjecture for Heisenberg isoperimetrices

Let $H(\mathbb R)$ be the Heisenberg group with a Carnot–Carathéodory metric arising from a norm on the horizontal plane. In the sub-Riemannian case, Pansu's bubble set is the unique surface, up to translation and dilation, having a Legendrian foliation by geodesics; it encloses a region and has constant mean curvature. Pansu's conjecture. The bubble set solves the sub-Riemannian isoperimetric problem: for every surface $S$ enclosing a region $E$,

$$\frac{\lambda^{3/4}(E)}{\operatorname{Sa}(S)}\leq \frac{3^{3/4}}{4\sqrt{\pi}}=0.3215\ldots,$$

with equality if and only if $S$ is, up to dilation and translation, the Pansu bubble set. This is the sub-Riemannian predecessor of the paper's generalized conjecture for arbitrary sub-Finsler metrics. The supplied status evidence says only that Pansu conjectured the bubble set as the solution, so the resolution status is not established by that evidence.