Algebraicity-or-density dichotomy for the Hodge locus

Let ${\mathbb V}$ be a polarizable $\mathbb{Z}$-variation of Hodge structures on an irreducible smooth quasi-projective variety $S$. Write $\operatorname{HL}(S, {\mathbb V}^{\otimes})$ for the Hodge locus and $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ for its typical part.

Algebraicity-or-density conjecture. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is empty, then $\operatorname{HL}(S, {\mathbb V}^{\otimes})$ is algebraic; otherwise $\operatorname{HL}(S, {\mathbb V}^{\otimes})$ is analytically dense in $S$.

The source states this as an immediate consequence of the two preceding conjectures, so it is a restatement of their predicted alternatives rather than an independent assertion. Its resolution status is not supplied.