Bounded critical number conjecture for $I_s$-free triangle-free matroids

Let $s\geq 4$. An $I_s$-free matroid is a binary matroid with no restriction isomorphic to the $s$-dimensional projective geometry $I_s$, and a triangle-free matroid has no three-element circuit. The critical number of a matroid $M=(E,G)$ is

$$\chi(M)=\dim(G)-\omega(M^c),$$

where $M^c$ is the complementary matroid and $\omega(M)$ is the dimension of a largest subgeometry of $M$ contained in $E$.

Bounded critical number conjecture. For any $s\geq 4$, the class of $I_s$-free and triangle-free matroids has bounded critical number.

The conjecture asks for a bound on the critical numbers depending only on $s$, not on the dimension of the matroids. The source notes that the result proved there settles the case $s=4$; the cases $s>4$ remain open in the supplied context.