There is no proof that COVID immunity is even possible at this point, the only way to test it would be to expose someone again to the virus, which is unethical.
Some parts of the world (Europe) are looking at new lock-downs as second and third waves of infection spread across their populations. Researchers should be looking to those countries and will most likely know already who had contracted Covid during the first wave/lock-down (the population themselves should know) which will enable researchers to study second and third infections by previous patients.
(Should it come to that)