The hypertight-contact-structure conjecture for manifolds with taut foliations

Let $V$ be a closed, orientable three-dimensional manifold. A codimension-one foliation on $V$ is taut if it is taut in the usual sense, and a contact structure is hypertight if it admits a Reeb vector field with no contractible periodic orbit.

Hypertight-contact-structure conjecture. Every closed, orientable manifold that carries a taut foliation also carries a hypertight contact structure.

This conjecture proposes a contact-geometric counterpart to the role of taut foliations in three-manifold topology. The source presents it as a more plausible expectation after discussing the relationship between hypertight contact structures, taut foliations, and Reeb dynamics; no resolution is given.