Symplecticity conjecture for transverse hyperplane sections of nearly Kähler embeddings

Let $X$ be the symplectic manifold and, for each positive integer $k$, let $F_k$ be the projective embedding constructed in the paper. A hyperplane section of $F_k$ is called transverse when the corresponding hyperplane intersects the embedded image transversely, so its inverse image is a smooth submanifold of $X$.

Symplecticity conjecture. For sufficiently large $k$, the transverse hyperplane sections of $F_k$ are symplectic submanifolds of $X$.

This would connect the nearly symplectic projective embeddings $F_k$ with the construction of symplectic submanifolds representing multiples of the symplectic class. The surrounding discussion presents this as a goal related to Donaldson's theorem, but the supplied text does not establish the assertion or give enough evidence to determine whether it has been resolved.