The structural conjecture for vertically 4-connected matroids with a modular 4-point line

Let $M$ be a simple, vertically $4$-connected matroid with a modular $4$-point line $L$.

Structural conjecture. Either $M$ is ternary, for some $e\in L$ the deletion $M\backslash e$ is binary, or $M$ has an $F_7$-minor and either an $F_7^-$- or $(F_7^-)^*$-minor.

This conjecture proposes that, among simple vertically $4$-connected matroids with a modular $4$-point line, the known ternary case and the case obtained by deleting one element to obtain a binary matroid account for all possibilities apart from the specified $F_7$ and non-Fano minor configuration. The surrounding discussion presents it as a likely classification and does not state that it has been resolved.