The explicit connectivity conjecture for breadth-critical tangles

Define a sequence of non-negative integers by $t_0=0$ and $t_i=3t_{i-1}+1$ for $i\geq 1$, equivalently $t_i=(3^i-1)/2$. Let $M$ be a matroid with a breadth-critical tangle $\mathcal{T}$ of order at least $k$. A matroid is $(t_0,t_1,\ldots,t_{k-2})$-connected if, whenever $F\subseteq E(M)$ has $\lambda(F)=i$ for $i\in\{0,1,\ldots,k-2\}$, either $|F|\leq t_i$ or $|E(M)-F|\leq t_i$. The explicit connectivity conjecture. If $\mathcal{T}$ is a breadth-critical tangle of order at least $k$ in $M$, then $M$ is $(t_0,t_1,\ldots,t_{k-2})$-connected. This is presented as a stronger version of the bounded connectivity conjecture, motivated by the bound $13=(3\times4)+1$ for $4$-separating sets in weakly $4$-connected matroids; it remains open.