Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I mean the derived dg category of quasicoherent sheaves on $X$) for some prestack $X$?

Or, perhaps by removing some of the conditions above, e.g. being $\text{QCoh}(X)$, $k$ having characteristic zero, stability.