I know there exists a groupoid-valued prestack parameterizing (connected) triangulated dg-categories (ie the points of this moduli are *not* objects of a fixed dg-category, but rather dg-categories themselves).

Has anything been done for stable $\infty$-categories?

[Connectedness is a condition on Hochschild cohomology: $HH^{<0}=0$ and $HH^0=k$.]