Apartment wrongfully asking to pay 10k$ in damages. What are my options?

Hello Everyone!

I moved out of an apartment July last year. 3 months before moving out, the A/C on the wall started leaking water from behind. I raised a work order request with the apartment asking them to fix it. The A/C eventually fell off the wall due to the leakage damaging the glue I guess? And it broke. Throughout this time , the apartment did not respond to the request and the lease eventually came to completion. The apartment then billed me 10k$ in damages in the move out statement due to the broken A/C. I contacted them saying that a work order request was already raised to fix the A/C eventually, but they denied even receiving such request. I had an auto generated email from the apartment that I got from the apartment on the day I raised the request. Unfortunately, the email does not contain any specifics to the request itself, just that they have received a work order request on xxx date. I lost access to the resident’s portal as well so I couldn’t take screenshot of the work order details

I sent them the email and a picture of the A/C dangling on the wall from one side due to the leakage along with a timestamp showing that it was 3 months prior to the move out date(the same picture was used in the work order request) . They did not respond to this email and sent the matter to their accounting department. When the accounting department contacted me, I sent them an email detailing how the damage took place due to management’s own negligence and asked them to look into the work order request for further proof. They responded saying that they have asked the management and they denied receiving any request so the conclusion of my complaint was that I am liable to pay. I asked them if they can share the work order request with me which was raised if they are denying that it was about the A/C. They have not responded to me yet but I’m pretty sure they are going to say that I’m liable to pay regardless. They have earlier told multiple times that the next step would be to just send the amount to collections which will impact my credit score. The apartment also is avoiding any written communications. Apparently it’s against their “policy” . So far all the correspondence has been over calls.

Is there anything I can do to protect myself from paying the 10k$? All I have is the picture of A/C dangling on the wall from one side with the timestamp and an email of work order request confirmation from the apartment.