Countability conjecture for decomposable transcendental Hodge structures of surface families

Let $f: \mathcal{S} \to B$ be a smooth projective family of complex surfaces over an irreducible base. For a fiber $S_b$ of $f$, let $\mathcal{T}_{S_b}$ denote its transcendental polarized Hodge structure.

Countability conjecture. There are at most countably many integral polarized Hodge structures $\mathcal{T}_{S_b}$, arising from the fibers $S_b$ of $f$, which are decomposable.

This conjecture is intended to imply the irrationality of a very general cubic fourfold: decomposable transcendental Hodge structures arising from surfaces would occur in only countably many cases. The supplied text gives no resolution of the conjecture.