Flatness comparison for rigid algebras over arbitrary bases

Determine whether, for an arbitrary presentably symmetric monoidal infinity-category \(\mathcal V\), a rigid \(\mathcal V\)-algebra \(\mathcal R\), and a dualizable \(\mathcal R\)-module \(\mathcal M\), an object is \(\mathcal V\)-flat if and only if it is \(\mathcal R\)-flat.

Background

The paper proves the equivalence between base-relative flatness and algebra-relative flatness for rigid algebras in the specific setting developed there. It explicitly asks whether the same comparison remains valid for arbitrary bases.

References

Let $\inCAlg()$ be arbitrary. Let $\inCAlg{rig}()$ be a rigid -algebra and be a dualizable -module. Is it true that an object $x\in$ is -flat if and only if it is -flat?

— Generalized Telescope Conjecture  (2609.03375 - Liang, 3 Sep 2026) in Section 7, subsection “Serre smashing frames versus pure frames”