Gompf's conjecture on Brieskorn spheres bounding Stein domains

A Brieskorn sphere is a Brieskorn homology sphere arising from the corresponding Brieskorn construction. A Stein domain in $\mathbb{C}^2$ is a compact domain with Stein structure whose boundary is the associated contact-type hypersurface.

Gompf's conjecture. No nontrivial Brieskorn sphere, with either orientation, arises as the boundary of a Stein domain in $\mathbb{C}^2$.

This conjecture addresses which Brieskorn spheres can occur as boundaries of Stein domains in the standard complex two-space. The authors state that their results leave it open and discuss only partial evidence.