Quasi-coherence conjecture for diagonal and Hochschild homology sheaves

Let $(X,\mathcal{C}^\infty)$ be a homologically regular reduced differentiable space. Let $\mathscr{E}_\bullet(\mathcal{C}^\infty_X)$ be its diagonal sheaf complex and let $\mathscr{H}\mathscr{H}_\bullet(\mathcal{C}^\infty_X)$ be its Hochschild homology sheaf. Quasi-coherence conjecture. The diagonal sheaf complex and the Hochschild homology sheaf are quasi-coherent.

If true, this would provide a sheaf-theoretic structural property for the complexes governing Hochschild homology on singular differentiable spaces. The supplied text gives no evidence that the conjecture has been resolved.