Large-girth minor conjecture for arbitrary cosimple matroids

Let $U_{2,q}$ be the $q$-element rank-$2$ uniform matroid, and more generally let $U_{t,q}$ be the $q$-element rank-$t$ uniform matroid. Let $M(K_t)$ be the graphic matroid of the complete graph $K_t$, and let $M(K_t)^*$ be its dual. Large-girth minor conjecture. For every positive integer $t$, there exists an integer $p(t)$ such that every cosimple matroid with girth at least $p(t)$ contains at least one of

$$U_{2,t+2},\quad U_{t,t+2},\quad M(K_t),\quad M(K_t)^*$$

as a minor.

The conjecture seeks a finite list of unavoidable minors for arbitrary cosimple matroids of sufficiently large girth, extending the representable case. The source presents it as a natural conjecture, with no resolution supplied.