Conjecture: ∃ℝ-hardness for maximum-degree-4 penny graph recognition

Prove that recognizing penny graphs remains complete for the existential theory of the reals when restricted to input graphs of maximum degree 4.

Background

Building on their hardness construction (which can be adapted to avoid degree-6 vertices), the authors suggest that ∃ℝ-hardness should persist even when the input graph has maximum degree 4, although this is not yet established.