The LP-E6 will power the grip, but does it actually charge them? It didn't with the grip I rented, but that was almost a year ago.
I kept on looking after reading your comment, since I expected that there was something. I did find a product advisory stating it would not charge from the grip. I have several LP-E6 batteries, but didn't know specifically which are in the grip. I just checked and they are both the N version, so maybe the old one won't charge. In any event, here is what I found from 1.5 years ago before the firmware update.
https://support.usa.canon.com/kb/index?page=content&id=ART173597&cat=0901e024801101e8&actp=LIST
The charging is sometimes weird, When I checked just now, one was at 69%, the other at 98% and was not charging when I powered up the camera with my 3rd party PD charger. Pulling out the battery tray and reinserting it started it charging. I might try connecting the Canon power supply to see if it works differently. I think its out in my studio.
It does always get charged, but I don't think a battery should fall below 75% before it starts.