Question about Covid-19 antibody testing re: people who suspect they had CV weeks/months a

Do we know how long Covid-19 antibodies stay in the system of someone who had it and recovered weeks or months ago before there was a test for it?

It would seem that if there was a solid reliable test for people who think they may have had CV-19 in say January it would make a huge difference in determining a more accurate mortality rate.

I realize this is strictly anecdotal buy I've heard from at least one prominent nurse that is pretty convinced the CV-19 virus has been circulating in the U.S. since AT LEAST early January if not as far back as early December. Many many cases here in the Mobile area of people going to the doctor and in some cases the hospital with some of the typical flu or bad cold symptoms in addition to unusually strong upper respiratory symptoms and less overall congestion including dry unproductive cough who tested negative for the flu.

Knowing if many of these people had actually had Covid-19 already would seem to be incredibly useful information in our response to the current crisis. If there is a fairly solid reliable antibody test for the virus and the antibodies for the virus would still be in someone's system a few months after recovering then I'm not sure why testing for this isn't an higher priority.