Pencil-like subgraphs in planar unit-copies

Characterize the bipartite graphs, beyond hypercubes, such that every planar unit-copy contains a large pencil-like subgraph.

Background

A pencil-like subgraph is a subgraph whose vertices are close to uniformly spaced points on a line; the paper uses such structures to construct planar colorings avoiding large hypercubes.

References

Which bipartite graphs, other than hypercubes, contain a large pencil-like subgraph in every unit-copy in the plane?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question, Section 6.1 (Planar case)