If there are no drugs involved, then there is no reason California Chrome shouldn't have the nasal strips. Apparently these are used to help horses with breathing issues.
I was under the initial impression these involve drugs, but they do not. They aren't any worse than tongue ties or blinkers or shadow rolls. ALL of those types of equipment help horses race but don't give them an unfair advantage.
If they did, Secretariat would have been regarded as a cheat because he always wore blinkers.
Video from a manufacturer showing how these strips work: