Generalised spike-structure conjecture for matroids with the $(t_1,2t_1,t_2,2t_2)$-property

Let a matroid have the $(t_1,\ell_1,t_2,\ell_2)$-property if every $t_1$-element set is contained in an $\ell_1$-element circuit and every $t_2$-element set is contained in an $\ell_2$-element cocircuit. Let $t_1$ and $t_2$ be positive integers.

Generalised spike-structure conjecture. There exists a function $f(t_1,t_2)$ such that if $M$ is a matroid with the $(t_1,2t_1,t_2,2t_2)$-property and

$$|E(M)| \ge f(t_1,t_2),$$

then $E(M)$ has a partition into pairs such that the union of any $t_1$ pairs is a circuit, and the union of any $t_2$ pairs is a cocircuit.

This conjecture generalises the structural conclusion for sufficiently large matroids with the $(t,2t)$-property. It is motivated by examples such as spikes and by results showing that certain sufficiently large matroids with related asymmetric properties form structured classes; the source does not indicate that this conjecture has been resolved.