By foundational work of Deligne, the Borel-Moore homology of the moduli stack of semistable sheaves on a smooth projective K3 surface carries a mixed Hodge structure. In general, this stack is singular and its structure morphism to a point is far from projective. Nevertheless, Halpern-Leistner has conjectured this Hodge structure to be pure. I will report on a work in progress with Ben Davison and sketch a proof of this conjecture, and its analogues for moduli in other CY2 categories, via non-commutative Donaldson-Thomas theory.