The universal tightness conjecture for contact structures with nowhere-invariant complex structure

Let $(M^3,\xi)$ be a closed contact manifold admitting a contact form $\alpha$ and a complex structure $J$ on $\xi$ such that

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

everywhere, where $X_\alpha$ is the Reeb field of $\alpha$. Universal tightness conjecture. Then $(M,\xi)$ is universally tight. This conjecture links the absence of Reeb-invariant compatible complex structures to tightness. The paper presents it after proving a related result for critical compatible metrics, but gives no resolution of the conjecture.