Purity conjecture for Borel–Moore homology of algebraic-symplectic stacks

Let $X$ be a variety, $P$ a polynomial, and $H$ a stability condition. Write $\mathcal{C}oh(X)_P^{H-\operatorname{ss}}$ for the moduli stack of $H$-semistable coherent sheaves on $X$ with Hilbert polynomial $P$. Let $\mathcal{X}$ be an algebraic-symplectic derived stack, and write $H_i^{BM}(\mathcal{X})$ and $H^i_{BM}(\mathcal{X})$ for its Borel–Moore homology and cohomology.

Purity conjecture. The Hodge structure on $H_i^{BM}(\mathcal{C}oh(X)_P^{H-\operatorname{ss}})$ is pure for any $H$, and, in general, $H_i^{BM}(\mathcal{X})$ is pure for any algebraic-symplectic derived stack $\mathcal{X}$ which has a proper good moduli space. More ambitiously, $H^i_{BM}(\mathcal{X})$ should be pure for any algebraic-symplectic derived stack $\mathcal{X}$ which is cohomologically proper.

The conjecture would give Hodge-theoretic control of Borel–Moore homology and support decompositions arising from derived Kirwan-type results. Purity is known for moduli of representations of preprojective algebras of quivers, but the stated generalizations remain open.