Universal-building category conjecture for semisimple Hitchin systems

Let $G$ be a semisimple complex Lie group, let $X$ be a Riemann surface, and let

$$\phi\in\bigoplus_{i=1}^{r}H^0(X,\omega_X^{\otimes d_i})$$

be a point in the corresponding Hitchin base. Let $h^{\phi}$ be the harmonic map to the universal building associated to $\phi$. Universal-building category conjecture. The singularities of $h^{\phi}$ determine a $3$-dimensional Calabi--Yau category $\mathcal C$. The paper presents this as an expected extension from the $\operatorname{SL}_3(\mathbb C)$ case to arbitrary semisimple Lie groups. It gives initial evidence but no general construction or proof.