Extension of the nilspace-character characterization to all orders

Prove that every $k$-step nilspace character is a character of order $k$ with quantitative complexity bounds depending on the precision parameter but not on the cardinality of the finite abelian group.

Background

The paper proves the correspondence between 2-step nilspace characters and quadratic characters, which is the key case needed for its algorithmic results. It explicitly anticipates the analogous statement at arbitrary order, but does not establish it.

References

We expect this latter fact to extend to general order $k$, but as explained in later sections, proving this is outside the scope of this paper.

— Spectral algorithms in higher-order Fourier analysis  (2501.12287 - Candela et al., 21 Jan 2025) in Remark following Lemma 3.5, Section 3.2