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We don't have consensus on the session's last block, so we shouldn't check if the cosign is before the session ends. What matters is that network, within its set, claims it's still active at that block (on its view of the blockchain).
Coordinator
The Serai coordinator communicates with other coordinators to prepare batches for Serai and sign transactions.
In order to achieve consensus over gossip, and order certain events, a micro-blockchain is instantiated.