General extremal-function conjecture for matroid classes over prime powers

For a prime power $q$, let $\mathcal{AC}_q$, $\mathcal{SL}_q$, and $\mathcal{AF}_q$ be the corresponding classes of matroids, with extremal functions $h_{\mathcal{AC}_q}(r)$, $h_{\mathcal{SL}_q}(r)$, and $h_{\mathcal{AF}_q}(r)$. Let $T_r^{q-2}$ be the indicated template and let $\varepsilon(T_r^{q-2})$ denote its size. General extremal-function conjecture. If $q$ is a prime power, then

$$h_{\mathcal{AC}_q}(r)\approx h_{\mathcal{SL}_q}(r)\approx h_{\mathcal{AF}_q}(r)\approx\binom{r+1}{2}+(q-2)(r-1)=\varepsilon(T_r^{q-2}).$$

This conjecture extrapolates the known extremal formulas for the regular, near-regular, and related matroid classes. The approximation relation and its precise scope are not further resolved in the supplied text.