The bounded connectivity conjecture for breadth-critical tangles

Let $k\geq 2$ and let $M$ be a matroid with a breadth-critical tangle $\mathcal{T}$ of order at least $k$. A matroid is $(s_0,s_1,\ldots,s_t)$-connected if, whenever $F\subseteq E(M)$ has $\lambda(F)=i$ for $i\in\{0,1,\ldots,t\}$, either $|F|\leq s_i$ or $|E(M)-F|\leq s_i$. The bounded connectivity conjecture. There is an infinite sequence $(s_0,s_1,s_2,\ldots)$ such that, for all $k\geq 2$, if $\mathcal{T}$ is a breadth-critical tangle of order at least $k$ in $M$, then $M$ is $(s_0,s_1,\ldots,s_{k-2})$-connected. This would extend the established fact that breadth-critical tangles of order at least $4$ occur in weakly $4$-connected, equivalently $(0,1,4)$-connected, matroids; the existence of a uniform sequence for all orders remains open.