The simultaneous breadth-preserving minor conjecture for tangle truncations

Let $M$ be a matroid with a tangle $\mathcal{T}$ of order $k\geq 4$. For each $i\in\{4,5,\ldots,k\}$, let $T_i(\mathcal{T})$ be the truncation of $\mathcal{T}$ to order $i$. The simultaneous breadth-preserving minor conjecture. There is a $(0,1,4)$-connected minor $N$ of $M$ such that, for all $i\in\{4,5,\ldots,k\}$, the tangle $T_i(\mathcal{T})$ generates a tangle $T'_i(\mathcal{T})$ in $N$, and

$$\operatorname{breadth}_M(T_i(\mathcal{T}))=\operatorname{breadth}_N(T'_i(\mathcal{T})).$$

This asks for one weakly $4$-connected minor that simultaneously preserves the breadths of all truncations of a tangle; the paper leaves this simultaneous reduction open.