Corti–Filip–Petracci smoothing-component conjecture

Let $X_0$ be an affine Gorenstein toric variety defined by the cone over a polytope $\triangle$ placed at height $1$. Let $\operatorname{Def}(X_0)$ be its versal $\mathbb{Q}$-Gorenstein deformation space, and call a connected component a smoothing component if it contains a variety with only terminal singularities. A zero mutable Laurent polynomial is a Laurent polynomial whose facet restrictions mutate to $1$, and its Newton polytope is the convex hull of its exponent vectors.

Corti–Filip–Petracci smoothing-component conjecture. Smoothing components of $\operatorname{Def}(X_0)$ are in bijection with zero mutable Laurent polynomials $f$ with Newton polytope $\triangle$.

In dimension three, the varieties in smoothing components are smooth because there are no Gorenstein terminal quotient singularities. The conjecture specializes the broader smoothing-type correspondence to a fixed affine toric variety and is not resolved in general by the supplied text.