Wood is strongest, or able to accept the greatest load without noticeable distortion, along the long axis of the grain. Conversely, wood will flex or distort most easily when force is applied perpendicular to the long axis of the grain. Cue shafts are cut along the long axis of the grain.
Gravity is a relentless enemy of wood.
If your cue is fully supported along its entire length, I suppose it would be better to store it horizontally. However, unless the tubes of your case are perfectly formed to the curvature of your cue, then it is not fully supported.
Also, it is important to understand the powerful effect of changing moisture content on wood.
Custom woodworkers, cabinet and furniture builders, etc, are familiar with the work of Dr. Gene Wengert. Do a search for him is you want to really understand the properties of wood.
Joe