One possibility is that each dungeon "tags" any person who enters it, by some magical mechanism. The final dungeon becomes available to someone who collects all of the "tags". It doesn't work if a bunch of different people collect some of the "tags" each.
The spell description for Magic Mouth is fairly specific about what it can't distinguish. The original AD&D version gave an example of how it could be used, and specified that it could distinguish age: "Speak only when an octogenarian female carrying a sack of groat clusters sits crosslegged within five feet." That example was removed at some point. But it has never been clear how the spell did its sensing, so it might be possible for a Magic Mouth to be triggered in the presence of "someone who has been through each of the dungeons". And it's possible to hide a dungeon behind a Magic Mouth -- for the opened lips to be a gateway. Or at least that's how I've used it, way back when.
There's a possibility of a "back door", easy to access, ridiculously implausible to find. "Visit the following sequence of eight doors, in order, without interruption" would do it.
See what others have posted; but also, the "ordeal" version used to be spelled differently: "gantlet" rather than "gauntlet".