BKU atypical Hodge locus conjecture

Let $S$ be an irreducible smooth quasi-projective algebraic variety, let $\mathbb V$ be a $\mathbb Z$-PVHS on $S$, and let

$$\Phi:S\to M=\Gamma\backslash\mathcal D$$

be its period map. Let $\mathrm{HL}(S,\mathbb V)_{\mathrm{atyp}}$ denote the union of atypical special subvarieties of $S$.

BKU atypical Hodge locus conjecture. The following conditions are equivalent: $\mathrm{HL}(S,\mathbb V)_{\mathrm{atyp}}$ is a finite union of maximal atypical special subvarieties; it is a proper algebraic subvariety of $S$; and it is not Zariski dense in $S$.

The source attributes this conjecture to the work cited as BKU and does not state a resolution.