>>19793336They use KaraFun's database when searching for a backing track, and this is the first step where they might be stopped when looking for a song. The three ways around are either providing the backing track yourself (partly why Mumei wanted a guitar), commissioning one, or singing the song acapella.
If they have an appropriate backing track, they then cross-reference that with a private document that says whether or not they have permission to sing it. This is the second step where they might get stopped, and as they generally don't comment on permissions more than having them or not, I assume it's up to their managers to negotiate the permissions.
Pic partially related -
https://twitter.com/rrin_reen/status/1442811039792254980 