Extension-of-scalars tensor structure for finite purely inseparable extensions

Prove or disprove that, for every artinian tensor category $C$ over a field $\Bbbk$ and every finite purely inseparable field extension $K/\Bbbk$, the scalar-extension category $C_K$ is a tensor category over $K$.

Background

The paper reduces the general scalar-extension problem to finite purely inseparable field extensions. A positive answer is known in several cases, including super-tannakian, semisimple, and finite tensor categories.

The authors isolate the general finite purely inseparable case as an unresolved question because rigidity of CKC_K, equivalently absolute flatness of KK as an algebra in CC, is not established in general.

References

It then follows quickly, see for instance \S 5, that answering Question~\ref{mainquestion} only requires considering finite purely inseparable extensions $K:\Bbbk$. \begin{question}\label{q:fields} Let $K:\Bbbk$ be a finite purely inseparable field extension and $C$ an artinian tensor category over $\Bbbk$. Is $C_K$ a tensor category? \end{question}

— Absolutely flat algebras in tensor categories  (2608.13137 - Coulembier et al., 13 Aug 2026) in Question \ref{q:fields}, Section 3.2, “Field extensions”