I suppose the investigators would look for inconsistencies in 'the story' as told by different parties. If 'the story' is true then additional detail will be forthcoming and generally non-contradictory between accounts. If its all a cover-up then individual suspects will invent different details and give themselves away. 'We know at least one of you is lying.' At which point someone will betray the conspiracy for favourable terms, and the police will have a real witness. Maybe also this is the reason for a 'captain's log' - if that went missing too then there'd be even more suspicion.