  In this work we continue to investigate well-posedness for stable driven
McKean-Vlasov SDEs with distributional interaction kernel following the
approach introduced in [8]. We specifically focus on the impact of the Besov
smoothness of the initial condition and quantify how it affects the
corresponding density estimates for the SDE. In particular, we manage to attain
some critical thresholds allowing to revisit/address in a stable noise setting
some concrete physical and biological models.
