Tightness conjecture for nowhere Reeb-invariant compatible metrics

Let $(M,\xi)$ be a contact 3-manifold. Let $X_\alpha$ be a Reeb vector field and $J$ a complex structure on the contact distribution, and let $\theta'$ denote the instantaneous rotation of a compatible metric. Tightness conjecture. If

$$\mathcal{L}_{X_\alpha}J\neq 0$$

everywhere, equivalently if there is a compatible metric with

$$Ricci(X_\alpha)<\frac{\theta'^2}{2}$$

everywhere, then $(M,\xi)$ is tight. The conjecture proposes a contact-topological obstruction to globally realizing this curvature and Reeb-dynamical behavior, and is presented as a generalization of earlier results on conformally Anosov contact 3-manifolds.