score:14
From this website http://www.iranicaonline.org/articles/clothing-ii , in a section concerning historical clothing it goes into length on several styles of headgear. This style falls under the designation 'tiara'.
In any event, the tiara had a top like a hood, often lined inside with luxurious animal fur. Ordinarily it was worn flat, either pressed down in front to form three knobs or falling in folds on either side. Only the great king had the right to wear his tiara (kyrbasΓa) βupright,β that is, with the top erect, presumably held by inner retainers (Xenophon, Anabasis 2.5.23; Arrian, Anabasis 3.25.3; Plutarch, Artaxerxes 26, 28; idem, Themistocles 29). (emphasis mine)
from the mosaic:
and a normal 'tiara' from the above mentioned web page shown in 'flat' style:
Upvote:2
It looks like a padded coif:
(source: tripod.com)
OT aside: you might also find this Youtube video from Lindeybeige on mail coifs interesting.