Conjecture on embeddings of graphs in $\mathcal{F}_{xy}^k$

Let $k\geq 1$ and let $G$ be a graph in $\mathcal{F}_{xy}^k$, the class of two-terminal graphs associated with alternating embeddings on the surface $\mathbb{S}_k$. Then $G$ embeds in $\mathbb{S}_k$.

Embedding conjecture. Every graph $G\in\mathcal{F}_{xy}^k$ embeds in $\mathbb{S}_k$.

The preceding results show that if such a graph does not embed in $\mathbb{S}_k$, then it is an obstruction for $\mathbb{S}_k$ and has an $xy$-alternating embedding in $\mathbb{S}_{k+1}$. The authors describe this exceptional scenario as quite unlikely; whether it can occur remains open.