Rigid triangle-free graphs in the plane

Characterize the triangle-free graphs that are rigid as unit-distance graphs in the plane, and determine whether every such graph $H$ satisfies $^2 \not\xrightarrow{2} H$.

Background

The paper discusses graphs whose planar unit-copies are sufficiently rigid to be avoided by staircase-like 2-colorings, and asks whether this phenomenon extends beyond the examples treated.

References

Which triangle-free graphs $H$ are ``rigid'' as unit-distance graphs in the plane? Is it true that $2 \not\xrightarrow{2} H$ for such graphs?

— 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)