Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a functor $X:(\mathbf{Aff}^{\mathrm{cn}})^{\mathrm{op}}\to \mathcal{S}$ satisfying fpqc descent. Then we can define its $\infty$-category of quasicoherent sheaves formally by a Kan extension.

Say that a symmetric monoidal stable $\infty$-category $\mathcal{C}$ 'has enough perfect objects' if its full subcategory of dualizable objects is dense (that is to say the induced functor $\mathcal{C}\to \operatorname{Ind}(\operatorname{Perf}(\mathcal{C}))$ is fully faithful).

Are there examples of fpqc stacks $X$ as above for which $\operatorname{QCoh}(X)$ does not have enough perfect objects?

What about if we restrict our question to ((Quasi)-Geometric Stacks, Artin Stacks, Deligne-Mumford Stacks, Algebraic Spaces, Schemes)?

For certain, this is true for quasicompact quasiseparated schemes and algebraic spaces, as well as quasi-Geometric stacks $X$ such that $\operatorname{QCoh}(X)$ is compactly generated and the structure sheaf is a compact object (proven in Lurie, Spectral Algebraic Geometry).

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