I first read this post yesterday (Wednesday) morning and having nothing constructive to add passed it buy I went to lunch that day at a Chinese Restaurant where a huge mural was displayed above the bar of a group of ancient Chinese Warriors, mounted in line with their Bows in Scabards. The Scabards were hung from the saddle on the left, with the Bows strung, battle ready, the quivers of arrows were slung to the right hand side of all of the riders in the mural.
The Bows were all in scabbards which were at the outside of the left leg of the rider and at least half of the length of the Bow was exposed.
For whatever this may be worth, and understand that the Chinese have a universal disdain for the worth of an individual, at some point in time this seems to have been an acceptable means of carrying a bow while mounted on a Horse.