The smooth regularity conjecture for \f-isoperimetric sets

Let $\phi$ be a norm of class $\mathrm{C}^\infty_+$, and let $\phi$-isoperimetric sets be isoperimetric sets for the norm $\phi$ in $\mathbb{H}^1$. Smooth regularity conjecture. Every $\phi$-isoperimetric set is of class $\mathrm{C}^2$. The conjecture concerns regularity at characteristic points, where the preceding results do not apply to general non-differentiable norms; it is presented as a condition needed for the subsequent result, and no resolution is given here.