Symplectic/contact analogue of the Looijenga conjecture

Let a cusp singularity be a surface singularity whose link carries the contact structure associated with its cusp resolution, and let a rational dual mean that the dual cusp singularity is rational. A Stein filling is negative definite when its intersection form is negative definite. Symplectic/contact analogue of the Looijenga conjecture. If a cusp singularity does not have a rational dual, then it admits only negative definite Stein fillings.

Looijenga's conjecture states that a cusp singularity is smoothable if and only if it has a rational dual. When the dual is rational, smoothing the cusp and resolving its dual produces a rational surface and gives a Stein filling with $b^+=1$; the asserted restriction in the non-rational-dual case remains open.