But what about the horizontal placement?
Two papers by Christopher W. Tyler address the "hypothesis of a consistent positioning of one eye relative to the center of the portrait" in a way I haven't seen done before.
AN EYE-PLACEMENT PRINCIPLE IN 500 YEARS OF PORTRAITS
Eye Placement Principles In Portraits And Figure Studies Over The Past Two Millennia
"Is there a consistent placement of the eyes relative to the canvas frame, based on the horizontal position of the eyes in portraits? Data from portraits over the past 2000 years quantify that one eye is centered with a standard deviation of less than + 5%. Classical texts on composition do not seem to mention the idea that the eyes as such should be positioned relative to the frame of the picture; the typical emphasis is on the placement of centers of mass in the frame or relative to the vanishing point in cases of central perspective. If such a compositional principle is not discussed in art analysis, it seems that its manifestation throughout the centuries and varieties of artistic styles (including the extreme styles of the 20th century) must be guided by unconscious perceptual processes."