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Existence of fcd stacks without perfect generation

Determine whether there exist quasi-geometric stacks of finite cohomological dimension whose category of quasi-coherent sheaves is not compactly generated; equivalently, decide whether there are fcd stacks that do not satisfy perfect generation.

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Background

For quasi-geometric stacks, perfect generation means QCoh is compactly generated and every perfect complex is compact.

In characteristic zero with affine stabilizers many stacks are fcd, but it remains unknown whether fcd can fail to imply perfect generation.

References

It still remains an open question whether there are examples of fcd stacks that do no satisfy perfect generation (Remark~9.4.0.1).

— $c$-structures and trace methods beyond connective rings (2509.14774 - Levy et al., 18 Sep 2025) in Example (Characteristic 0), Section 6.1.3 (Compact generation on stacks)