A conformal symplectic Weinstein conjecture

We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we call Reeb 2-curves. The conjecture readily holds for all closed exact lcs surfaces. In higher dimensions, we give partial verifications of this conjecture, based on certain extended ($\mathbb{Q} ^{} \sqcup \{\pm \infty\}$ valued) Gromov-Witten, elliptic curve counts in $\lcs$ manifolds. As a basic application we get some novel results in classical Reeb dynamics. The most basic such result gives sufficient conditions for a strict contactomorphism to fix the image of some closed Reeb orbit on a closed contact manifold. Along the way we give a Gromov-Witten theoretic construction of the classical dynamical Fuller index (for Reeb vector field), which among other things explains its rationality.