Not sure if VNL rules will be similar to Olympics, but unlike in v-league, you have to challenge a play immediately (and interrupt the play) if you notice something - unlike v-league where you can challenge any moment of the play at the very end.
Thought this rule was made pretty clear early in, but not sure why even in this last set, Korea was challenging an antenna touch (that happened mid-play) only after the play ended.
International volleyball's rules are pretty consistent (it's the same in VNL and OG), which is that you have to challenge within five seconds (edit: 8 seconds) after the fault happened. If it's something mid-rally and the ref doesn't call it it's the coach's responsibility to interrupt the play.
There is no way Lavarini doesn't know this. If his assistant is responsible for pressing the challenge button then he should know he's not in V-league.