Three-triangle matroid minor conjecture

A triangle in a matroid is a circuit of size $3$. A matroid admits another matroid as a minor if that minor is obtained by deletions and contractions. Three-triangle matroid minor conjecture. If every element of a matroid $\mathcal{M}$ is contained in $3$ triangles, then $\mathcal{M}$ has one of

$$\mathcal{M}(K_5),\qquad F_7,\qquad \mathcal{U}_{2,4}$$

as a minor. The source gives no resolution; the conjecture is motivated by the Fano matroid, the uniform matroid $\mathcal{U}_{2,4}$, and the graphical matroid of $K_5$ as basic examples with this property.