In this
POST there is a recent report from someone with a six hour connection who visited several lounges. There is no mention of the three hour rule in the post, but perhaps [MENTION=504332]AlphaTango[/MENTION] can tell us if they entered the Centurion Lounge before the three hour threshold.
When I entered the Centurion, it was definitely within 3 hours of my outbound from LAX. (But I would definitely expect to have been admitted had it been outside that window, as it was a connection. I've done this in the recent past and just had to show both boarding passes - one showing I'd just arrived and the other showing I was departing the same airport - that time was just over 3 hours later.)