Structural conjecture for 3-connected binary matroids excluding M(K_4) as an induced minor

Let a binary projective geometry be a projective geometry representable over $GF(2)$, and let a tipped coning mean a coning operation that retains its tip, while a tipless coning means deleting the tip from a coning. A matroid is triangle-free if it has no three-element circuits. Generalized parallel connections across projective geometries, tipped conings, and tipless conings of triangle-free matroids are the operations under consideration.

Structural conjecture. The class of $3$-connected binary matroids that do not contain $M(K_4)$ as an induced minor is exactly the class of matroids that can be obtained by starting with binary projective geometries and circuits and applying sequences of the following operations: generalized parallel connections across projective geometries, tipped coning, and tipless coning of triangle-free matroids.

This conjecture aims to characterize all $3$-connected binary matroids excluding $M(K_4)$ as an induced minor. The surrounding results establish closure properties for the relevant class under generalized parallel connections and coning operations, but the stated characterization remains to be proved.