Binary matroid conjecture excluding an independent set and a triangle

Let $s$ be a positive integer. A simple binary matroid is a binary matroid with no loops or parallel elements; an induced $I_s$-restriction and an induced $\operatorname{PG}(1,2)$-restriction are restrictions isomorphic to the indicated matroids. Write $\chi(M)$ for the chromatic number of a matroid $M$. Binary matroid chromatic-number conjecture. For every $s \ge 1$, there exists an integer $k$ such that if $M$ is a simple binary matroid with no induced $I_s$-restriction or $\operatorname{PG}(1,2)$-restriction, then

$$\chi(M) \le k.$$

This is the $t=2$ case of the proposed matroidal analogue of the Gyárfás–Sumner conjecture. The corresponding general matroidal statement is refuted already when $s=t=3$, while this triangle-excluding case remains open; its graph-theoretic analogue is known for stars but open in general.