The holomorphic Weinstein conjecture for characteristic foliations

Let $X$ be a compact holomorphic symplectic manifold and let $Y\subset X$ be a hypersurface of holomorphic contact type. The characteristic foliation on $Y$ is the rank-one foliation induced by the restriction of the symplectic form to $Y$.

Holomorphic Weinstein conjecture. The generic leaf of the characteristic foliation on $Y$ is a rational curve. In particular, if $Y$ is smooth, then it is a $\mathbb{P}^1$-bundle over the space of leaves $Y/F$.

This is proposed as the holomorphic analogue of the real Weinstein conjecture, in which a compact contact-type level set should contain a periodic Hamiltonian orbit. The context explains that a compact holomorphic leaf corresponding to a periodic orbit is a rational curve, but does not establish the conjecture.