The Q-smoothing conjecture for terminal Fano threefolds

Let $X$ be a Fano threefold with terminal singularities. A Q-smoothing of $X$ is a Q-Gorenstein family $\mathcal{X}\rightarrow C\ni 0$ such that

$$\mathcal{X}_0\cong X,$$

and $\mathcal{X}_t$ has quotient singularities for $t\neq 0$. The Q-smoothing conjecture. Every Fano threefold with terminal singularities has a Q-smoothing.

The preceding theorem establishes this for Fano threefolds with ordinary terminal singularities; the conjecture concerns the general terminal case, including exceptional non-ordinary singularities.