Since no one has answered, I'll just toss this out.
My vet/hunting partner did some tabletop toe surgery on a shorthair a few hunting seasons ago. The dog had a split nail that started growing back _inside_ the nail bed and was carrying that paw. Got all infected and swollen. The vet had to remove the nail, root and all, so it woudn't grow back then antibiotics. Dog was using the paw the next day but not hunting of course.
All I can say is keep an eye on it as it regrows and make sure it grows out normally because that was one ugly operation to watch.
I have 2 of the pieces off now. Third one is resilient!! I had some anti biotics on hand so I had already started those along with cleaning twice a day. The soft part of the nail is growing and getting harder. No swelling and acts like nothing is wrong. I still am trying to get that 3rd piece and treating it like a loose tooth as I keep moving it back and forth as much as he will let me anyway. It is all I can do to hang on to him when I start pulling on it!