From Official Wiki of the Renaissance Kingdoms

A conspiracy allows a Head of the Clan position to be taken over by force, without removing the current leader from the family clan.

When a Clan Member has more influence points than the current clan leader, they can launch a conspiracy to take their place.

In order for a conspiracy to succeed, at least 25% of the clan members must join the conspiracy within 7 days, so that the sum of the influence points of the various conspirators reaches at least 40% of the influence points of the family clan. At the end of the 7 days, the conspiracy fails if the required conditions are not met.

If two conspiracies succeed at the same time (passage of day), the conspiracy whose initiator has the most influence points succeeds. The other conspiracy is cancelled.

If a character with a shared connection initiates or joins a conspiracy, then the other characters sharing their connection cannot join the same conspiracy.