It is authorized by linking the root of the MAST m in P and verifying the script is indeed a leaf of the MAST via Merkle proof.
During setup, all parties jointly sign all possible surrogate scripts.
Instead of committing four scripts in a MAST in a single transaction tx2, each script can be made into a separate transaction.
In the uncooperative case, one of these four transactions, containing the chosen script, can be broadcasted.
Since Graftroot on Bitcoin does not require any consensus changes, unlike on BTC, it can be leveraged today for improved privacy and efficiency.