Nonvanishing conjecture for Hochschild homology of admissible subcategories

Let $X$ be a smooth projective variety, let $\D^b(X)$ denote its bounded derived category of coherent sheaves, and let $\CA\subset \D^b(X)$ be an admissible subcategory. Write $\HOH_\bullet(\CA)$ for the Hochschild homology of $\CA$. Nonvanishing conjecture. If

$$\HOH_\bullet(\CA)=0,$$

then

$$\CA=0.$$

The conjecture asserts that every nonzero admissible subcategory of the derived category of a smooth projective variety has nonvanishing Hochschild homology. The supplied text does not state a resolution, so its status remains open.