You might try applying one of these to the doors and other surfaces that surround the AC. Note that the second is for cars but a commenter used it to reduce the noise through his entry door so I guess it can be used for other purposes. I'd remove the cabinet doors first so you can attach the material to horizontal surfaces where possible.

• https://www.amazon.com/Loaded-Soundproofing-Barrier-Highest-Quality/dp/B01EGX3XF0
• https://www.amazon.com/Noico-deadening-Automotive-Insulation-dampening/dp/B01KZ5X7KO

-- amateur

If that air conditioner is running and you have to have those cabinet doors open, I would want a nice front on the ac unit. You should have gotten one with the AC unit when you bought it. It isnt pretty to look at the way it is. Maybe time to call the people that installed it, because, it really should not be that loud either.

The through-the-wall AC certainly came with a front but the owner isn't going to be able to attach it because some duct work was added to the upper-front of the unit to direct the conditioned air upward (there's a grill in the cabinet top in front of the window).

If I were the owner I would check whether the unit should operate in a closed cabinet. If not, I'd cut holes in the doors of the required size and cover them with perforated metal (on the inside) to allow air flow and eliminate the need to leave the doors open. To reduce noise, I'd also put the noise-reducing material on the floor below the AC unit, behind the toekick grill.

-- not an AC guy

That whole setup is designed to burn up the AC as quickly as possible. A couple of holes just isn’t going to do it. The whole surround thing needs to go away.