The $S(K_3)$-free signed bipartite planar graph conjecture

Let $(G,\sigma)$ be a signed bipartite planar graph, let $S(K_3)$ be the signed subdivision of $K_3$, and write $(G,\sigma)\to(K_{3,3},M)$ for the corresponding signed homomorphism. $S(K_3)$-free signed bipartite planar graph conjecture. If

$$S(K_3)\not\to (G,\sigma),$$

then

$$\chi_c(G,\sigma)\leq 3,$$

i.e., $(G,\sigma)\to(K_{3,3},M)$. This is proposed as a potential strengthening of the paper's main result that would include Grötzsch's theorem as a special case; it is presented among the paper's further open questions.