ag.algebraic geometry – Deformation equivalent Hodge structures

An HH type is the oriented homotopy type of a closed simply-connected Kähler manifold together with the Hodge structure on cohomology.

Two HH types are deformation equivalent if they are realized by closed Kähler manifolds that are deformation equivalent.

If two HH types are equivalent as oriented homotopy types and have the same Hodge diamonds are they deformation equivalent?

Closed simply-connected complex surfaces with $p_gneq0$ (e.g. complete intersections) can supply counterexamples. I haven’t yet verified if the statement holds for them.