Planar Lefschetz fibrations and Stein structures with distinct Ozsváth-Szabó invariants on corks

Thanks to a result of Lisca and Matić and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsváth-Szabó invariants. Here we give an infinite family of examples showing that converse of Lisca-Matić-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.