I'm sure osage sapwood is tougher than most white woods heartwood, however I would remove it. Why eat hotdogs when you can have steak? Properly chasing a single back ring will produce a pristine back that will last a lifetime. No offense to you hotdogs lovers, I'm just sayin...lol Danny
I wonder if it really is better than white woods. I have only small experience with osage, and always chase a ring, but I once found a 4" plus black locust sapling, ramrod straight and clean as a whistle, with NO heartwood. I made a bow that would have worked beautifully for elm, floor tillered it, heat treated the belly, and got it on the tree. It was so stiff for the dimensions. But when I got it bending at all (with a scale at 50 lbs) the wood just sat down, crumpled and folded up. No elasticity whatsoever. Stiffness, but no return.
The bow I was making would have been fine in maple, elm, white mulberry, honey locust, etc...and wide than needed in black locust heartwood.
Either way, danny, I'd chase a ring on osage.