Finite unavoidable-minor conjecture for large-girth matroids

Let $B(G)$ denote the bicircular matroid of a graph $G$, let $U_{t,t+2}$ be the uniform matroid of rank $t$ on $t+2$ elements, and let $M(K_t)$ and $M(K_t)^*$ be respectively the graphic matroid of the complete graph $K_t$ and its dual. Finite unavoidable-minor conjecture. There exists a function $g$ such that for every integer $t$, every cosimple matroid with girth at least $g(t)$ contains one of

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

as a minor.

This conjecture would imply the preceding finite-characterization conjecture, adding bicircular matroids to the natural minor-minimal classes with arbitrarily large girth. The source gives no resolution.