Thank you @captaincook. Very clear information and understandable for users not yet familiar with the intricacies of obj/mtl files.
(Merci @captaincook. Des informations très claires et compréhensibles pour les utilisateurs qui ne sont pas encore familiarisés avec les subtilités des fichiers obj/mtl.)

Here's a MirrorShpere for other users to download.

@Andrei Maybe the Ni and Ns don't make a difference. At least the Ni shouldn't without transparency. Did you try different values that didn't appear to make a difference?
I think the most important part is the sweethome3d_window_mirror prefix in the OBJ file. That might overrule some values in the MTL file for the photo renderer.