Simpson's motivicity conjecture for perverse sheaves and constructible complexes

Let $X$ be a complex algebraic variety. Let $\C_X$ be $Perv(X,\mathbb{C})$ or $\mathbf{D}^b_c(X,\mathbb{C})$, and let $S=\{\mathcal{F}\}\subset\operatorname{Iso}(\C_X)$ consist of one object. A unispace structure on $\C_X$ gives meanings to $K$-constructible, $K$-closed, and absolute sets. Simpson's generalized motivicity conjecture. The following assertions should hold: (1) $S$ is $K$-constructible if $\mathcal{F}$ is defined over a subfield $K$ of $\mathbb{C}$; (2) $S$ is $K$-closed if $\mathcal{F}$ is a semisimple perverse sheaf defined over $K$; and, if $X$ is smooth, (3) $S$ is absolute $\mathbb{C}$-constructible; (4) if $\mathcal{F}$ is a semisimple perverse sheaf, then $S$ is absolute $\mathbb{C}$-closed; (5) $S$ is absolute $\mathbb{Q}$-closed if and only if $\mathcal{F}$ is of mixed-Hodge-module origin; (6) $S$ is absolute $\overline{\mathbb{Q}}$-closed if and only if $\mathcal{F}$ is of geometric origin. The difficult parts are (5) and (6); the earlier parts are expected to be approachable using moduli constructions, but the conjecture as a whole remains open.