Klingler's characterization conjecture for variations of Hodge structures with dense CM points

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$, with generic Hodge datum $({\pmb G},{\mathcal D})$. A point of $S$ is a CM point for $(S,{\mathbb V})$ if it has the corresponding CM Hodge-theoretic property. Let ${\rm Sh}_K({\pmb G},{\mathcal D})$ be the Shimura variety associated with the datum, let ${\rm Sh}_K^\circ({\pmb G},{\mathcal D})$ be a connected component, and let ${\mathbb V}_\rho$ be the standard ${\mathbb Z}$-variation associated with an algebraic representation $\rho:{\pmb G}\to \operatorname{GL}(V)$. Klingler's conjecture. If the set of CM points for $(S,{\mathbb V})$ is Zariski-dense in $S$, then $({\pmb G},{\mathcal D})$ is a Shimura datum and there is a Cartesian diagram

\begin{tikzcd} {\mathbb V}=\psi^*{\mathbb V}_\rho \arrow[d] \arrow[r] & {\mathbb V}_\rho \arrow[d] \\ S \arrow[r,"\psi"] & {\rm Sh}_K({\pmb G},{\mathcal D}) \end{tikzcd}

where $\psi$ is dominant onto ${\rm Sh}_K^\circ({\pmb G},{\mathcal D})$. This is a Hodge-theoretic form of the André–Oort principle: dense CM points should force the variation to arise from a Shimura variety. The source presents this as Klingler's Conjecture 5.3 and gives no resolution status.