Stability-condition component conjecture for Hitchin categories

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

$$\mathcal C$$

be the $3$-dimensional Calabi--Yau category associated to the singularities of the harmonic map to the universal building, and let

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

be the corresponding Hitchin base. Stability-condition component conjecture. There exists a component of the moduli space of stability conditions of $\mathcal C$ which contains $B$. The source further expects this component to contain the base of the Hitchin system, but gives no resolution of the stated existence claim.