Conjecture on Euler impure graphs on Dyck's surface

Let $\mathbb{N}_3$ denote Dyck's surface, and let an Euler impure graph be an edge-maximal graph embeddable on a surface that does not triangulate that surface. For a graph $G$ and a graph $H$, write $E(H)$ for the edge set of $H$; thus $K_8-E(H)$ is obtained from $K_8$ by deleting the edges of $H$. Conjecture on Euler impure graphs on Dyck's surface. The graphs

$$K_8-E(2K_2) \quad\text{and}\quad K_8-E(K_{1,2})$$

are the two unique Euler impure graphs on Dyck's surface $\mathbb{N}_3$. The conjecture proposes a finite characterization of the Euler impure graphs on the remaining non-orientable surface of genus three. The source presents it as a tractable conjectural characterization, and the supplied text gives no evidence that it has been resolved.