Yes we do (I have to dig out the reference), but it appears to be the same as other Coronaviruses.
You are right that the worse the infection the longer the immunity, but if we are looking at the young they have a mild disease so we should expect a short immune lifetime.
Edit. Have a look at Figure 3 of this pre-print[0]. This doesn’t encourage me that we can expect a long immune lifetime with SARS-CoV-2.
0. https://www.medrxiv.org/content/10.1101/2020.04.15.20066407v...
