Brettell–Chun–Goddyn–Whittle connectivity conjecture for matroids

Let $t$ be a positive integer and let $M$ be a matroid. A matroid is $(2t-1)$-connected in the usual matroid-connectivity sense, and $M\backslash X$ and $M/X$ denote deletion and contraction of a set $X$, respectively. An $(s,t)$-spike is a matroid of the corresponding spike type.

Brettell–Chun–Goddyn–Whittle conjecture. There exists a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that if $M$ is a $(2t-1)$-connected matroid with no circuits or cocircuits of size $2t-1$, and

$$|E(M)| \ge f(t),$$

then either there exists a $t$-element set $X \subseteq E(M)$ such that either $M/X$ or $M\backslash X$ is $(t+1)$-connected, or $M$ is a $(t,t)$-spike.

This conjecture was stated by Brettell, Chun, Goddyn, and Whittle; the case $t=2$ was proved by Williams. The statement is motivated by matroid-connectivity results characterizing spikes as highly connected matroids without small circuits or cocircuits, but its general case is not resolved in the supplied source.