Holomorphic Seifert–Weinstein conjecture for locally conformally symplectic manifolds

Let $S^{2k+1}\times S^1$ be the product of an odd-dimensional sphere with a circle, and let $C$ be a threefold. An l.c.s. pair is a pair $(\omega,J)$ consisting of a locally conformally symplectic form $\omega$ and a compatible almost-complex structure $J$.

Holomorphic Seifert–Weinstein conjecture. For any l.c.s. pair $(\omega,J)$ on $S^{2k+1}\times S^1$ or on $C\times S^1$ for $C$ a threefold, there is an elliptic, non-constant $J$-holomorphic curve.

The claim proposes removing the delta-nearby hypothesis from the preceding existence theorem. It is motivated by the existence of Reeb orbits, and hence Reeb tori, in the standard odd-dimensional sphere and for contact structures on threefolds, but the general assertion is left as a conjecture.