Invariant Reeb orbit conjecture for contactomorphisms

Let $(Y,\xi)$ be a closed contact manifold equipped with a selected contact form $\alpha$ such that $\ker\alpha=\xi$. Let $\phi$ be a contactomorphism of $Y$ that preserves $\alpha$ and is contact-isotopic to the identity. A $\phi$-invariant Reeb orbit is a Reeb trajectory whose image is shifted by $\phi$ by some nonzero time. Invariant Reeb orbit conjecture. Every such $\phi$ admits a $\phi$-invariant Reeb orbit. This is presented as a generalization of the Weinstein conjecture; the supplied text gives no resolution of the claim.