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A 2-dimensional open book (S, h) determines a closed, oriented 3-manifold Y (S,h) and a contact structure ξ (S,h) on Y (S,h) . The contact structure ξ (S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse holds, that is On the other hand, results by Wand  and by Baker, Etnyre and Van Horn-Morris  imply the existence of counterexamples to ( * ) with S of arbitrary genusdoi:10.2140/agt.2014.14.531 fatcat:dvlfikerwzftfdy64grksb42hi