A 5-chromatic same-distance graph in the hyperbolic plane

The chromatic number of the plane problem asks for the minimum number of colors so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the answer is 5, 6, or 7. Here we consider the problem in the context of the hyperbolic plane. We prove that there exists a distance $d\approx 1.375033509$ so that every 4-coloring of the hyperbolic plane contains two points distance $d$ apart, which are identically colored.