The torsion conjecture for admissible subcategories with vanishing Hochschild homology

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $\mathcal{A}\subset\mathcal{D}^b(X)$ be an admissible subcategory, meaning that its inclusion has both left and right adjoints. Write $HH_{\bullet}(\mathcal{A})$ for its Hochschild homology and $K_0(\mathcal{A})$ for its Grothendieck group. The torsion conjecture. If

$$HH_{\bullet}(\mathcal{A})=0,$$

then $K_0(\mathcal{A})$ is a torsion group. This is presented as a consequence expected from standard motivic conjectures; the source gives no proof or resolution.