Symplectic foliated fillings of sphere cotangent bundles

We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.