Surface bound conjecture for signed graph genus

Let $S$ be a surface, let $n_S$ be the largest order of a complete graph that embeds into $S$, and let $\gamma(G,\sigma)$ and $\gamma^*(G,\sigma)$ denote the relevant orientable and non-orientable genera of a signed graph. Surface bound conjecture. If $S$ is not the projective plane, then

$$\gamma(G,\sigma) \leq \gamma(\pm K_S) \quad\text{and}\quad \gamma^*(G,\sigma) \leq \gamma^*(\pm K_S),$$

and there are graphs for which equality holds. The statement is presented as a conjecture, with a theorem identified as a first result toward its proof; the supplied text gives no resolution status.