Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

A question in low-dimensional symplectic topology asks whether every compact contractible 4-manifold admits the structure of a Stein domain. A related conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in C2, with either orientation. We give the first known example of a contractible, boundary-irreducible 4-manifold that admits no Stein structure with either orientation, though its boundary has Stein fillings with both orientations. We also verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in C2. A byproduct of our example is the construction of a cork that does not admit a Stein structure with either orientation.