Equivalent strong Zilber–Pink conditions for the atypical Hodge locus

Let $S$ be an irreducible smooth quasi-projective variety endowed with a polarizable variation of Hodge structures ${\mathbb V}\to S$. Let $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{atyp}}$ denote the atypical Hodge locus, and let an irreducible subvariety be optimal for ${\mathbb V}$ in the sense used by the source.

Strong Zilber–Pink equivalence. The following conditions are equivalent: (a) the atypical Hodge locus is a finite union of maximal atypical special subvarieties; (b) it is a strict algebraic subvariety of $S$; (c) it is not Zariski-dense in $S$; and (d) $S$ contains only finitely many irreducible subvarieties optimal for ${\mathbb V}$.

The source presents this as an enhanced strong-form Zilber–Pink conjecture, implying the formulation in the cited earlier work. The candidate has no supplied resolution evidence.