Looijenga's smoothability conjecture for cusp singularities

Let $(X,x)$ be a cusp singularity, and let its dual cusp be the cusp obtained by dualizing the configuration of curves in its minimal resolution. Say that the minimal resolution of the dual cusp sits on a rational surface if it occurs as the anticanonical cycle in an anticanonical pair $(Y,D)$, where $Y$ is a smooth rational surface and $D\in |-K_Y|$ is a cycle of rational curves of the same type as the minimal resolution of the dual cusp.

Looijenga's conjecture. The cusp $(X,x)$ is smoothable if and only if the minimal resolution of the dual cusp sits on a rational surface.

Looijenga proved that sitting on a rational surface is necessary for smoothability; the conjecture asserts that it is also sufficient.