Bu then t nothing else happens, that is, tuning doesn't actually engages, until I send a second midi PC#125 (which corresponds to ACTIVATE?).

Some manufacturers show MIDI program change numbers starting at 1 while other show them starting at 0. In the H9 they start at 0. So, perhaps, the device you're using starts program change messages at 1, and your 125 corresponds to Eventide's 124?