The signed-graph $K_5$-minor conjecture for maps to signed projective cubes

Let $k$ be a positive integer, let $(G,\sigma)$ be a signed graph, let $(K_5,-)$ denote $K_5$ with all edges negative, and let $\mathcal{O}_{k+1}$ be the set of signed cycles that do not map to $C_{-(k+1)}$. A signed graph has neither a $(K_5,-)$-minor nor an induced element of $\mathcal{O}_{k+1}$ when both conditions hold.

Signed-graph $K_5$-minor conjecture. Any signed graph $(G,\sigma)$ which has neither a $(K_5,-)$-minor nor induces an element of $\mathcal{O}_{k+1}$ maps to $SPC(k)$.

This is presented as a deeper extension of the preceding conjecture, replacing ordinary minor-freeness by a signed-graph minor condition and extending the known characterization for signed partial $3$-trees.