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.



Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated?
If $X$ is a locally compact topological space, then
$\mathrm{Shv}(X, \mathrm{Mod}\_{k} )$ is a presentable $k$-linear stable $\infty$-category which is dualizable (in fact, self-dual), but is rarely compactly generated (for example, this fails for $X = \mathbf{R}$).


