Tian–Todorov-type smoothness problem for $X_\zeta$ when $\Gamma\subset\operatorname{SU}(3)$

Let $\Gamma\subset\operatorname{SU}(3)$, let $X_\zeta$ be the moduli space associated with a generic parameter $\zeta$, and let $(M^{0,*,\Gamma},\bar{\partial}_\alpha)$ be the differential graded Lie algebra governing its deformations at $\alpha\in\mathcal N^\Gamma\cap\mu^{-1}(\zeta)$. Its Kuranishi germ is the germ describing the local deformation space at $\alpha$. Tian–Todorov smoothness problem. Can one imitate the Tian–Todorov proof and show that, for generic $\zeta$, the Kuranishi germ of $(M^{0,*,\Gamma},\bar{\partial}_\alpha)$ is isomorphic to ${\mathcal H}^{0,1,\Gamma}_\alpha$, equivalently that $X_\zeta$ is smooth? The proposed analogy is with the Tian–Todorov argument for smoothness of Calabi–Yau deformation spaces; the question remains open in the stated setting.