David is right, the horn tip should have been fixed with animal glue. I have soaked these in solvent and they have stayed put but since they are usually easily removed using nothing but patience and warm water, it may be sensible to remove the tip and re-fix afterwards. No need to use animal glue then, epoxy works much better.
The only snag is if someone has re-fixed the tip with PVA (white glue). That could give you a headache! It MIGHT dissolve in the solvent, no way of telling until you try.