Determine normability of all regular cardinals

Determine whether every regular cardinal is $T$-normable for an atomic orbital category $T$, rather than merely $T$-restrictable, in the sense that the corresponding small-category subcategories are closed under the relevant indexed norm functors.

Background

The paper distinguishes TT-restrictability, which controls closure under parametrised restrictions and pullbacks, from NN-normability, which additionally requires closure under indexed norms. In the atomic orbital case, the authors state that every regular cardinal is TT-restrictable, but they do not know whether all regular cardinals are TT-normable. This matters for the generality in which the small and presentable normed-category constructions apply.

References

It is, however, not clear that $\kappa$ is always $T$-normable, and we suspect that this is not the case.

— Normed idempotence of spectra and distributivity in equivariant higher algebra  (2610.22074 - Hilman et al., 15 Oct 2026) in Section 1, subsection “Norms of presentable categories,” Example following Theorem 3.6