Bonamy–Kardoš–Kelly–Nelson–Postle conjecture on bounded critical number

A simple binary matroid is a restriction of a finite binary projective geometry $\operatorname{PG}(n-1,2)$. It is $I_s$-free if it has no $s$-element independent flat, and its critical number is $n-k$, where $k$ is the dimension of a largest subgeometry of $\operatorname{PG}(n-1,2)$ disjoint from the matroid. A matroid is triangle-free if it has no triangle restriction.

Bonamy–Kardoš–Kelly–Nelson–Postle conjecture. For any $s\geq 1$, the simple $I_s$-free and triangle-free binary matroids have bounded critical number.

This conjecture is analogous to the Gyárfás–Sumner conjecture for graphs. It is trivial for $s=1,2,3$ and was resolved by the authors for $s=4$, but remains open for $s\geq 5$. The analogous assertion fails when triangle-free is replaced by $F_7$-free, because of the even-plane matroids.