Absolute-set decomposition conjecture for local systems

Let $X$ be a smooth complex algebraic variety, and let $\mathcal{M}_B(X)$ be the moduli space of semisimple local systems of all ranks. An absolute $K$-constructible set is a subset satisfying the paper's absolute constructibility conditions over $K$. Absolute-set decomposition conjecture. Every absolute $\mathbb{Q}$-constructible subset of $\mathcal{M}_B(X)(\mathbb{C})$ contains a $\mathbb{C}$-local system of mixed-Hodge-module origin; consequently, $\mathcal{M}_B(X)(\mathbb{C})$ is the smallest closed absolute $\mathbb{Q}$-constructible set containing all such local systems. Every absolute $\overline{\mathbb{Q}}$-constructible subset contains a $\mathbb{C}$-local system of geometric origin; consequently, $\mathcal{M}_B(X)(\mathbb{C})$ is the smallest closed absolute $\overline{\mathbb{Q}}$-constructible set containing all local systems of geometric origin. This is part of the paper's conjectural picture connecting absolute sets with the decomposition theorem; its assertions remain open.