When I switch off a zone, the damper of that zone opens

If the damper of a zone opens when said zone is switched off, it could be the case that the polarity of the actuator's output is not correct. (Red / +) and (Black / -). Check it.

If the polarity is correct, check whether the actuator operates properly ( The actuators of my dampers do not operate ).

Take into account that in the Airzone Easyzone system the AC unit is switched off when all the zones are switched off and that all dampers are opened in four minutes ( The dampers open when I switch off all my system's zones ).