The conjecture on the spectrum of growth rate functions for exponentially dense matroid classes

Let $q$ be a prime power, and let $\mathcal{M}$ be a base-$q$ exponentially dense minor-closed class of matroids, meaning that its growth rate is eventually of order $q^n$. Let $h_{\mathcal{M}}(n)$ denote the maximum size of a simple rank-$n$ matroid in $\mathcal{M}$. Growth-rate spectrum conjecture. There exist integers $k$ and $d$ with $k \ge 0$ and

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

such that

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

for all sufficiently large $n$. This conjecture proposes that the displayed values are the extremes in a small spectrum of possible eventual growth rate functions; the source further conjectures that every allowable triple $(q,k,d)$ occurs for some minor-closed class.