Ok, thanks. I did not know that.
Also, from the JP wiki I see there will be an exchange option for the higher tier cores. 1 of them costs 3 lower tier cores and 6 of the regalia fragments, 3 times per day. So that is another way to get more cores to grind the raid, although it seems like a pretty bad deal. Although I guess if you are MVP then it will net cost you 13 fragments for a regalia, and also spawn the raid that can get your friends fragments also.