Critical-number strengthening for dense triangle-free binary matroids

Let $M$ be a simple triangle-free binary matroid, with ground-set size $|M|$ and rank $r(M)$. Critical-number strengthening conjecture. If

$$|M|>\tfrac14 2^{r(M)},$$

then $\chi(M)\in\{1,2\}$. The preceding theorem gives only bounded critical number above this density threshold; the conjecture predicts the sharper possibilities, by analogy with the corresponding graph result.