There are many myths and anecdotal stories about what woods can be attacked by wood-boring insects and how they should be protected. In fact, wood-boring insects are but one of hundreds of life forms that will feed or live within organic material like wood and therefore you should become aware of the kinds of biologicals that will degrade the quality of your staves. Basically, the list encompasses three major groups: insects, fungi, and bacteria. Oftentimes we'll concentrate on the insects only and then have our staves attacked by fungi or bacteria, both of which can degrade wood in a matter of weeks or months depending on where you live and the conditions underwhich you store your wood. So here's a simple rule to follow: Store your wood in an area with low humidity (to reduce fungus growth) and in an area where insects are unlikely to enter. Low humidity tends to also inhibit bacterial growth but not always. So let's refine this rule a bit more:
1) The more air-tight your storage room (not exposed to outside predation) the less likely your chances of being invaded by wood-boring insects. But it's also a good idea to fog the room at least once a month with an insecticide fogger. Pre-historic bowmakers did not store dozens of staves as some do now. Their obsessions perhaps leaned in other directions. But they protected their staves by keeping them in smoky areas. This kept the wood-boring insects away and also inhibited fungal growth.
2) Go to the store and buy yourself an ultra-violet light (or two depending on your room size) and keep it on whenever you are NOT in the room. The ultra-violet light will absolutely inhibit fungal growth. As long as you are going to go through all the effort to acquire your staves then you might as well take every precaution to save them.
3) Leaving any sort of wood out in the elements (and Osage orange is just another organic material that can be attacked) will surely begin a degradation process. Not too many years ago I performed an analysis of the types of fungi that had invaded some Osage orange fence posts that had been removed prior to "development" by a local builder. The posts were about 70-80 years old and though many were rotting there seemed enough that were suitable for bows. Insects had reeked havoc on the posts over the decades but the real culprit was fungal invasion. Though we were able to extract some very fine bows from some of these posts, we found only a few free of fungal invasion. If you look at organic material invaded by fungus under a microscope you'll be amazed at how disruptive it can be to the fibers and overall structural dynamics.
So why take the chance? Store in a safe, low humidity room and use the occasional fogger and don't forget the ultra-violet light. Why go through all the work and then not take every precaution possible?
Good luck.