The highly vertically connected matroid minor conjecture

Let $M$ be a matroid. For an integer $k$, $M$ is vertically $k$-connected if, for every $A\subseteq E(M)$ with $\lambda_M(A)<k-1$, either $A$ or $E(M)-A$ is spanning in $M$. Write $M(K_n)$ for the cycle matroid of the complete graph, $B(K_n)$ for its bicircular matroid, $U_{n,2n}$ for the uniform matroid, and $M^*$ for the dual matroid. Highly vertically connected matroid minor conjecture. For all $n\geq 2$ there is an integer $k$ such that, if $M$ is a vertically $k$-connected matroid with $|M|\geq 2k$, then $M$ or $M^*$ has a minor isomorphic to one of $M(K_n)$, $B(K_n)$, or $U_{n,2n}$. The conjecture predicts the unavoidable minors for highly vertically connected matroids in minor-closed classes omitting a uniform matroid; the dual outcomes reflect that the relevant graph and bicircular matroid duals can have highly vertically connected minors.