The mustache does not show in cycles at any exposure or color management setting. Displacement and bump maps of the greyscale image texture do not help at all. Bump mapping just adds a bunch of polygonal-looking dark spots all over the mesh, while displacement does nothing. Comparing cycles to Eevee, the original model looks like a different person as the eyebrows are unrealistically desaturated, the mustache is nowhere to be seen, and the skin on the face is mostly one shade of white.
The model was created by UV mapping two images of a person on the stock man mesh that was provided by a blender 3.6 package. Additional non-quad destroying sculpting was done in Eevee to get the head to match the image texture. I did it in Eevee because my computer isn't good enough to sculpt in cycles.