Winding-number collections for $K_5$ minus an edge

Determine which collections of winding numbers $w_f(C,v)$ are realizable by almost embeddings of the graph obtained from $K_5$ by deleting one edge.

Background

The general realizability problem is explicitly stated to be unresolved for the graph obtained from K5K_5 by deleting an edge. This graph is notable because a related integer constraint is known: for an almost embedding of K5K_5 with one edge removed, the difference between two specified winding numbers is always $\pm1.

The open issue is to determine the full set of winding-number collections, rather than only the single relation currently known.

References

The answer to the open problem in the following interesting cases is unknown: \begin{itemize} \item $K$ is obtained from $K_5$ by deleting an edge (cf. Theorem 1);

— On winding numbers of almost embeddings of $K_4$ in the plane  (2501.15642 - Alkin et al., 26 Jan 2025) in Section 5, immediately following the Open Problem