An asymptomatic infection should still count as an infection, so it should lower the infection fatality rate. I don't have expertise in this area, but I think both infected fatality rate and symptomatic fatality rate are useful.
For the study, they tested a representative sample that includes asymptomatic cases. Hence, one can assume a known, constant number of infected cases. And to complete the study they should, as far as I understand it, wait until they know how many of those infected die in order to calculate the fatality rate. I don’t think it’s relevant here if further people are infected after testing for the illness, and there shouldn’t be a need to monitor the new cases or even check if they die (it shouldn’t be necessary for the study).
