Conditioning two-party quantum teleportation within a three-party quantum channel

We consider an arbitrary continuous-variable three-party Gaussian quantum state which is used to perform quantum teleportation of a pure Gaussian state between two of the parties (Alice and Bob). In turn, the third party (Charlie) can condition the process by means of local operations and classical communication. We find the best measurement that Charlie can implement on his own mode that preserves the Gaussian character of the three-mode state and optimizes the teleportation fidelity between Alice and Bob.