Conjecture on the regularity and lightlikeness of Hitchin–Li–Mochizuki quasicircles

Let $h_{\mathbf{H}^2}$ be the Hitchin map and let $\sigma_{\mathbf{H}^2}$ be the Hitchin–Li–Mochizuki section, whose image parametrizes quasisymmetric maps associated with the corresponding maximal surfaces. Hitchin–Li–Mochizuki conjecture. The image of the Hitchin–Li–Mochizuki section consists of quasisymmetric maps whose image is $\mathcal{C}^1$ and lightlike everywhere. This predicts additional regularity and a geometric lightlike property for the quasicircles arising from the Hitchin–Li–Mochizuki construction; the supplied text does not indicate that the conjecture has been resolved.