Representability by canonical gain-graph incidence matrices

Let $(G,\mathcal{B})$ be a biased graph, let $M$ be an elementary lift of the frame matroid $F(G,\mathcal{B})$, and suppose that $M$ is $3$-connected. Let $\mathbb{F}$ be a field and let $A$ be a matrix representing $M$ over $\mathbb{F}$. Canonical incidence-matrix conjecture. There is an $(\mathbb{F}^+\rtimes\mathbb{F}^{\times})$-gain function $\psi$ for $G$ such that

$$F(G,\mathcal{B})=F(G,\psi/\mathbb{F}^+),$$ $$M=M(\mathbb{F}^+\rtimes\mathbb{F}^{\times},\mathbb{F}^+,\mathcal{A},G,\psi),$$

and $A$ is projectively equivalent to $A(D,\psi)$ for every orientation $D$ of $G$. This asserts that every such representation is, up to projective equivalence, an incidence matrix arising from a gain function.