If the asset quality review and the subsequent stress test reveal that banks need to be recapitalised, the need for capital will have occurred on the watch of the national supervisors – so you could say it is a legacy issue. It is therefore down to the member states in question to repair these legacy issues before responsibility shifts to the ECB.
Private funds should be first in line to address any capital shortfalls. Failing that, it should be up to the member state to recapitalise the bank in question, assuming the latter runs a viable business model. This asset quality review is a major undertaking for supervisors and the banks in question.