Switching-and-scaling equivalence conjecture for canonical gain-graph matrices

Let $\mathbb{F}$ be a field, let $G$ be a loopless graph with orientation $D$, and let $\psi$ and $\psi'$ be $(\mathbb{F}^+\rtimes \mathbb{F}^{\times})$-gain functions for $G$. Suppose that

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

and that this matroid is $3$-connected and has rank at least $3$. The gain functions are switching-and-scaling equivalent if one can be obtained from the other by a sequence of vertex switchings and scalings. Canonical matrix equivalence conjecture. The matrices $A(D,\psi)$ and $A(D,\psi')$ are projectively equivalent if and only if $\psi$ and $\psi'$ are switching-and-scaling equivalent. This would extend the known frame and lifted-graphic special cases; the loopless, connectivity, and rank hypotheses are stated to be necessary.