Non-fullness of higher-step free nilpotent Reidemeister spectra

Prove that the Reidemeister spectrum of N_{r,c} is not full for every pair of integers r≥2 and c≥3.

Background

The full Reidemeister spectrum is SpecR(N_{r,c})=N₀∪{∞}. The cited prior work establishes that the spectrum is never full when c≥r and conjectures non-fullness for all higher-step free nilpotent groups with c≥3, including the parameter range 3≤c<r that remains outside the established result.

References

They also show that \SpecR(N_{r, c}) is never full if c \geq r and conjecture the following: Let r \geq 2 and c \geq 3 be integers. Then the Reidemeister spectrum of N_{r, c} is not full.

— Reidemeister spectra of free nilpotent groups and plethysms of Schur functions  (2501.02591 - Senden, 5 Jan 2025) in Section 1, subsection “Starting point: Reidemeister spectra of free nilpotent groups”