Wall mounted AC:
According to our real room test, wall mounted AC wouldn’t affect much of your indoor air quality. So your purifier along, if it can clean your room efficiently, should work the same when you have the wall mounted AC on during hot summer of freezing winter.
Central air system:
Since the building manager sometimes gives the inaccurate info and some many things could go wrong, use a particle counter to test out whether it keeping bringing in outdoor air.
Central air filtration system:
The system is designed to clean your indoor air while bringing outdoor air in. Remember to test it regularly and change the filter as needed.
More details is in this article: https://smartairfilters.com/en/blog/does-ac-bring-in-pm2-5/