The maximum partial-dual genus bound for planar graphs

Let $G$ be a planar graph, let $G^c$ denote its complement, let $\chi(G^c)$ be the chromatic number of $G^c$, and let $y_G$ denote the maximum genus among partial duals of $G$. For an integer $\lambda$, say that $G$ is $\lambda$-edge-connected if every edge cut of $G$ has size at least $\lambda$.

Maximum partial-dual genus conjecture. If $G\not\cong C_3$ is a $\lambda$-edge-connected planar graph with $\lambda\in\{1,2\}$, then

$$y_G\leq 2\chi(G^c)-\lambda+1.$$

This conjecture proposes a chromatic-number bound for the maximum partial-dual genus in the cases of 1- and 2-edge-connected planar graphs, where the authors report that they have not found graphs attaining the previously obtained bound. The even paths and even cycles discussed immediately before the conjecture satisfy the proposed inequality.