The critical-number conjecture for dense triangle-free binary matroids

Let $M$ be a simple triangle-free binary matroid, let $r(M)$ denote its rank, and let $|M|$ denote its number of elements. The critical number of $M$ is the minimum codimension of a flat of the ambient projective geometry disjoint from $E(M)$. Dense triangle-free binary matroid conjecture. For each real number $\alpha>0$ there exists $c\in\mathbb Z$ such that, if

$$|M|\geq \alpha 2^{r(M)},$$

then $M$ has critical number at most $c$. The conjecture would extend the paper's main theorem from odd circuit lengths $k\geq 5$ to triangles; the graph analogue fails for triangle-free graphs, so the binary-matroid assertion is a distinct structural question.