Lance was in a group with Nate, Taako, Lup, Barry, and Kravitz, and they turned over their shard to the Lab. Otherwise, Lance hasn't been involved with that stuff; he's probably been more focused on shuffling Name around since the Lab's been busy and such.

Connor didn't have anything because fuck that, HOWEVER he has definitely been lurking around the Lab to keep an eye on things, or more specifically to look out for Alphys. He doesn't think having a whole bunch of magical artifacts that people have been getting all mad at each other about in one place with no security is a good idea.