Pansu's isoperimetric conjecture for the sub-Riemannian Heisenberg group

Let $H(\mathbb R)$ be the Heisenberg group with its sub-Riemannian Carnot–Carathéodory metric. Let $S$ be a surface enclosing a region $E$, let $\lambda$ denote Haar measure, and let $\operatorname{Sa}(S)$ denote sub-Riemannian surface area. The isoperimetric ratio is $\lambda^{3/4}(E)/\operatorname{Sa}(S)$. Pansu's conjecture. For every such surface,

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

with equality if and only if $S$ is, up to dilation and translation, the Pansu bubble set. Equivalently, the bubble set is the unique isoperimetrix. Pansu constructed the bubble set as the surface formed by geodesics from the origin to $(0,0,1)$; it is a topological sphere with a Legendrian foliation by geodesics and constant mean curvature. The conjecture asserts that it solves the sub-Riemannian isoperimetric problem. Its resolution status is unclear from the supplied source context.