Optimal dual-separable embeddings of complete graphs

Let $\delta_1(c)$ denote the minimum genus of a dual-separable embedding of a $c$-connected graph, and let $\gamma(K_{c+1})$ denote the genus of the complete graph $K_{c+1}$. Optimal dual-separable embedding conjecture. For $c \geq 8$,

$$\delta_1(c)=\gamma(K_{c+1})+2,$$

except when $c=11$. The paper establishes this for several small values of $c$ and for the infinite family $c=12s+4$ with $s\geq2$, but the case $c=11$ remains unresolved; the authors' evidence motivates the conjecture.