Existence of Reeb 2-curves in rational exact lcs manifolds

Let $M$ be a closed manifold of dimension at least $4$, and let $\omega=d^{\alpha}\lambda$ be an exact locally conformally symplectic form on $M$ whose Lee form $\alpha$ is rational. A Reeb 2-curve is a smooth map $u:\Sigma\to M$ from a closed, possibly nodal Riemann surface such that $u_*(T\Sigma)\subset\mathcal{V}_{\lambda}$ and there is a smooth map $o:S^1\to\Sigma$ with $o^*u^*\lambda(s)\neq0$ for every $s\in S^1$. Reeb 2-curve existence conjecture. There is a Reeb 2-curve in $M$. The conjecture is motivated by the fact that the paper's holomorphic-curve arguments produce Reeb 2-curves and then deduce Reeb curves; no resolution is given in the source.