A one-parameter family of symplectic forms on an orbifold, followed by resolution and torus blowup, produces a non-Kähler manifold with both HLC and non-HLC symplectic forms in the same connected component.
Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems,J
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Symplectic non-K\"ahler manifolds with and without the Hard Lefschetz Condition
A one-parameter family of symplectic forms on an orbifold, followed by resolution and torus blowup, produces a non-Kähler manifold with both HLC and non-HLC symplectic forms in the same connected component.