Realizability with a 2-tree short subgraph

Determine whether every bipartite dichotomous ordinal graph whose short subgraph is a 2-tree admits a geometric realization in the Euclidean plane.

Background

The paper proves planar realizability in the plane under several structural restrictions on the short or long subgraph, notably when the short subgraph is outerplanar. A 2-tree is a broader graph class than the specifically treated settings, and the paper does not establish whether this restriction on the short subgraph guarantees realizability.

The conclusion explicitly identifies the 2-tree case as an unresolved question for bipartite dichotomous ordinal graphs.