Generalized connectivity conjecture for matroids with circuits and cocircuits of different cardinalities

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

Generalized connectivity conjecture. There exists a function $f: \mathbb{N}^2 \rightarrow \mathbb{N}$ such that if $M$ has no circuits of size at most $2s-1$, no cocircuits of size at most $2t-1$, is $(2\min\{s,t\}-1)$-connected, and

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

then at least one of the following holds: there exists an $s$-element set $X \subseteq E(M)$ such that $M/X$ is $(s+1)$-connected; there exists a $t$-element set $X \subseteq E(M)$ such that $M\backslash X$ is $(t+1)$-connected; or $M$ is an $(s,t)$-spike.

This is proposed as a generalization of the earlier conjecture, whose symmetric case has circuits and cocircuits of the same forbidden size. The source notes that sufficiently large $(s,t)$-spikes provide the obstruction, but does not report a proof or disproof of the generalized claim.