Simpson's algebraicity conjecture for the non-abelian Hodge locus

Let ${\mathcal Y}\to {\mathcal S}$ be a smooth projective family over a quasiprojective base, and let $M_{\rm dR}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ and $M_{\rm Dol}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ be the relative de Rham and Dolbeault moduli spaces. Let $N_{\rm dR}$ be the image, under the non-abelian Hodge correspondence, of the fixed-point locus of the scaling action on the Dolbeault space, and let $M_{\rm dR}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n({\mathbb Z}))$ denote the flat bundles with integral monodromy. Define the non-abelian Hodge locus by

$$\mathrm{NHL}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n):=N_{\rm dR}\cap M_{\rm dR}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n({\mathbb Z})).$$

Simpson's conjecture. The non-abelian Hodge locus $\mathrm{NHL}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ is an algebraic variety, and its inclusions into $M_{\rm dR}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ and $M_{\rm Dol}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ are algebraic morphisms. This conjecture predicts algebraicity of the locus of flat bundles underlying integral polarizable variations of Hodge structure, together with compatibility of its de Rham and Dolbeault realizations; the source provides no resolution status for the conjecture.