The half-breadth internally 4-connected minor conjecture

Let $M$ be a matroid with a tangle $\mathcal{T}$ of order $k\geq 4$ and breadth $m$. The half-breadth internally 4-connected minor conjecture. The matroid $M$ has an internally $4$-connected minor $N$ with a tangle $\mathcal{T}'$ of order $k$ such that $\mathcal{T}$ generates $\mathcal{T}'$ in $N$ and the breadth of $\mathcal{T}'$ is at least $m/2$. This would strengthen the weakly $4$-connected outcome available when breadth is preserved exactly, by allowing a controlled loss of breadth; the conjecture is stated as an open problem in the paper.