Negotium Perambulans in Tenebris

I know Story card’s effects are able to be initated
by the player who won that story but are all
Conspiracy cards worded like Story cards?
Conspiracy cards effects indicate if
the effect is initiated upon winning
the conspiracy. If they do not indicate
the effect is optionally triggered upon
winning, then the effect is a passive
effect that is always active and resolves
as indicated on the card. NOTE:
Conspiracy cards are not considered to
have a controller, and any instance of
the word “you” or “your” refers to all
players.

Player whose turn it is wins it, I think

When Frozen time gets play, during the "resolve passive effects" part of that process, the win condition is re-checked to see if each player has won. The active player gets to go first in this kind of process, so if they have 5 or more tokens on the story at that point, they win it (regardless of how many any other player has, even if they had the same number or more - there is no "most tokens" criteria in story victories).