Bell twirling followed by triplet symmetrization

Average the four correlated local Pauli operations to remove coherences in the Bell states basis. A one-party Pauli operation can permute the largest of the four resulting weights into the singlet sector; that weight is at least $1/4$. Correlated common local unitaries then permute the three triplet projectors, while collective-unitary covariance of the two-qubit singlet keeps the singlet projector unchanged. Averaging all six permutations makes the triplet weights equal, producing a two-qubit Werner state. Shared randomness or local operations and classical communication coordinates these local operations. Common unitaries alone cannot increase an initially small singlet weight, because that weight is invariant under them.