Conjecture on the unique signed graph realizing $n_s(4,3)$

Let $n_s(\lambda,p)$ be the minimum number of vertices of a signed graph with balanced chromatic number at least $p$ and negative girth at least $\lambda$. Signed-graph extremal conjecture. One has

$$n_s(4,3)=13,$$

and the only signed graph with $13$ vertices, balanced chromatic number $3$, and negative girth $4$ is the signed graph depicted in the paper's Figure 13. This is the next open case after the known observation $n_s(3,p)=2p-1$; the supplied text gives no proof or resolution of the conjecture.