Altınok–Brown–Reid conjecture on smoothing Q-Fano threefolds

Let $X$ be a Q-Fano threefold, meaning a three-dimensional Fano variety with $Q$-Cartier canonical divisor. Let $\Delta^1$ be a unit disc, and let $D\in |{-}K_X|$ when such an element is chosen. A Q-smoothing is a deformation $f\colon\mathcal{X}\to\Delta^1$ whose general fiber has only quotient singularities; a simultaneous Q-smoothing of $(X,D)$ is a deformation $f\colon(\mathcal{X},\mathcal{D})\to\Delta^1$ of the pair such that the general fiber $\mathcal{X}_t$ has only quotient singularities and $\mathcal{D}_t\in|{-}K_{\mathcal{X}_t}|$ has only Du Val singularities on the singularities of $\mathcal{X}_t$. Altınok–Brown–Reid conjecture. (i) Every Q-Fano threefold $X$ admits a Q-smoothing. (ii) If $|{-}K_X|$ contains an element $D$, then $(X,D)$ admits a simultaneous Q-smoothing.