I think it is very logical to think HVAC is a major contributor to infections in nursing homes and I would say it borders on criminal neglect.
Unfortunately, some of the people in authority have turned a blind eye to many articles, reports, literature affirming coronavirus can be spread by air, and HVAC can spread the infection. Even the American Society of Heating, Refrigerating and Air-Conditioning Engineers ASHRAE has clearly stated that the virus can be spread by HVAC. They don't recommend turning off the HVAC with a flimsy excuse that the discomfort may be harmful to health. We are after all capitalists.
But just turning off the AC may not solve the problem completely. The airborne virus can travel from one room to another via the hallway through diffusion and air currents caused by opening and closing doors and people movement. The Vietnamese are very smart in this respect according to one source. They use only natural ventilation even in hospitals and have 1000 times lower infection rates USA vs Vietnam. In fact, all the poor undeveloped countries have orders of magnitude lower infection rates because they cannot afford HVAC. Their shopping places are primitive like the farmer's market and street hawkers. Attached are some typical pictures from the internet, showing how people shop and what I mean by open ventilation.
The nursing home management is very poor. Residents don't go out, so the attendants are probably bringing the infections in. I was unable to bring this up to the powers there are. There is so much arrogance there.