Basically the title of the question. For the sake of completeness I state an introduction to the question.

In “Right-veering diffeomorphisms of compact surfaces with boundary I” and II, the authors introduce the notion of right-veering diffeomorphism of a surface with boundary. They show that there is a gap between the group of automorphisms that are composition of right-handed Dehn twists $ Dehn^{+}(\Sigma)$ and the group of right-veering diffeomorphisms $ Veer(\Sigma)$ . That is, the content $ Dehn^+(\Sigma) \subset Veer(\Sigma)$ is strict.

In their first paper, they prove this gap by showing some examples of automorphisms in $ Veer(\Sigma) \setminus Dehn^+(\Sigma)$ :

(1) Monodromy maps for open book decompositions supporting tight      contact structures which are not holomorphically fillable. (2) Right-veering monodromy maps of open book decompositions      supporting overtwisted contact structures. 

In their second paper of the series they show extra examples of the punctured tours. In particular they show pseudo-Anosov diffeomorphisms that are right-veering but are not in $ Dehn^+(\Sigma)$ . And they also prove that for (freely) periodic automorphisms of the punctured torus there is no gap between the two groups.

My question is if it is known that the last holds for every surface with boundary (basically the title of the question) and if the answer is no, what are the counterexamples.