Hwang's algebraicity conjecture for characteristic foliations

Let $X$ be an irreducible holomorphic symplectic manifold and let $Y\subset X$ be a smooth hypersurface such that $\mathcal{O}(Y)$ is ample. The characteristic foliation on $Y$ is the rank-one foliation induced by the restriction of the symplectic form to $Y$.

Hwang's conjecture. The characteristic foliation on $Y$ cannot have all of its leaves algebraic.

This conjecture concerns the incompatibility between positivity of the hypersurface and algebraicity of every characteristic leaf. The supplied context notes that Hwang and Viehweg proved the stronger relevant result for smooth hypersurfaces of general type, which implies this conjecture.