Rank-preserving weak-map injection conjecture for chains of matroid flats

Let $\mathsf{M}$ and $\mathsf{N}$ be matroids on the same ground set, and suppose there is a rank-preserving weak map $\mathsf{N}\to\mathsf{M}$. Let $\Delta(\mathcal{L}(\mathsf{M}))$ and $\Delta(\mathcal{L}(\mathsf{N}))$ be the sets of all chains of flats of $\mathsf{M}$ and $\mathsf{N}$, respectively. Rank-preserving weak-map injection conjecture. There exists an injective map

$$\varphi\colon\Delta(\mathcal{L}(\mathsf{M}))\to\Delta(\mathcal{L}(\mathsf{N}))$$

that sends the empty chain to the empty chain and sends every chain $F_0\subsetneq\cdots\subsetneq F_m$ to a chain $G_0\subsetneq\cdots\subsetneq G_m$ satisfying $\operatorname{rk}_{\mathsf{M}}(F_i)=\operatorname{rk}_{\mathsf{N}}(G_i)$ for every $0\leq i\leq m$. The source reports that this former conjecture was proved by Miyata, Proudfoot, and the fourth author; it is therefore solved.