The decategorification isomorphism conjecture for Hochschild homology

Let ${\cat V}$ be a smooth and proper DG category over ${\mathbb{k}}$. There is an injective algebra homomorphism

$$\pi\colon \underline{H}_{\mathrm{HH}_\bullet({\cat V})} \hookrightarrow \mathrm{HH}_\bullet\left(\bicat{H}_{{\cat V}}\right).$$

Decategorification isomorphism conjecture. The injective algebra homomorphism above is always an isomorphism.

The conjecture asks whether the Heisenberg algebra constructed from the Hochschild homology of a smooth proper DG category exhausts the Hochschild homology of the associated categorified Heisenberg object. The source presents this as a conjectural strengthening of the proved injective decategorification map; no resolution is supplied here.