Even-circuit bounds for uniform-minor-free matroids

Prove or disprove that, for every fixed integer q≥2, every rank-n simple U_{2,q+2}-minor-free matroid with no circuits of size 2k has size O(q^{n/k}).

Background

The paper’s matroid theorem establishes the corresponding bound for simple F_q-representable matroids. The proposed extension replaces representability by the weaker condition of excluding U_{2,q+2} as a minor, a class that generally lacks the linear structure used in the proof.

References

For every fixed integer $q \geq 2$, is it true that every rank-$n$ simple $U_{2,q+2}$-minor-free matroid with no circuits of size $2k$ has size $O(q{n/k})$?

— Generic solutions to symmetric linear equations  (2610.02177 - Frederickson et al., 1 Oct 2026) in Question, Section 6, subsection “Matroids excluding an even circuit and a uniform minor”