Pre-Lagrangian tori transverse to an Anosov flow

An Anosov flow on a smooth three-manifold $M$ gives rise to a Liouville structure on $\mathbb{R} \times M$ by a construction of Mitsumatsu. In a recent paper, Cieliebak, Lazarev, Massoni and Moreno ask whether an embedded torus $Σ\subseteq M$ transverse to an Anosov flow gives rise to a Lagrangian in $\mathbb{R} \times M$. We show that the answer to this question is in general negative by finding a topological obstruction related to the foliations induced on the torus by the weak stable and unstable bundles of the flow. Going in the opposite direction, we show that the answer is positive if the induced foliations do not admit parallel compact leaves.