Main conjecture on atypical loci as finite unions of special subvarieties

Let $S$ be an irreducible smooth quasi-projective variety endowed with a variation of mixed Hodge structures $\mathbb{V} \longrightarrow S$. Denote by $S_{\mathrm{atyp}}(\mathbb{V})$ the union of all atypical subvarieties of $S$ for $(S,\mathbb{V})$. Main conjecture, form 1. The subset $S_{\mathrm{atyp}}(\mathbb{V})$ is a finite union of special subvarieties of $S$. This is the strongest of four equivalent formulations predicting that atypical loci, unlike general Hodge loci, have simple geometry; its status is not resolved in the source.