Eventual form conjecture for exponentially dense minor-closed matroid classes

Let $\mathcal{M}$ be a minor-closed class of matroids satisfying condition

$$of Theorem$$

for some prime power $q$.

Eventual form conjecture. There exist an integer $k\geq 0$ and an integer $d$ with

$$0\leq d\leq\frac{q^{2k}-1}{q^2-1}$$

such that

$$h_{\mathcal{M}}(n)=\frac{q^{n+k}-1}{q-1}-qd$$

for all sufficiently large $n$.

The paper presents this as a proposed generalisation beyond the $\operatorname{GF}(q^2)$-representable setting. It predicts that exponentially dense minor-closed classes have an eventually uniform growth-rate formula, but no resolution is supplied here.