Milnor–Tjurina conjecture for Q-Gorenstein smoothings

Let $(V,0)$ be a non-Gorenstein surface singularity admitting a $\mathbb Q$-Gorenstein smoothing, meaning a smoothing obtained as an $r$-cyclic quotient of a smoothing of the index-one cover. Milnor–Tjurina conjecture for Q-Gorenstein smoothings.

$$\mu\geq \tau-1,$$

with equality if and only if $(V,0)$ is quasi-homogeneous. Since the source notes that $\alpha_\pi=0$ for such smoothings, this is the corresponding conjectural bound for the difference between the Milnor and Tjurina numbers. It is verified there when the canonical cover is a hypersurface singularity, while the general case is left open.