People keep citing studies done in a handful of small Italian towns allegedly showing that ~60% of people had already been infected, but in a random sample of healthy blood donors in Milan, prevalence of immunity was only ~7% by the beginning of April. medrxiv.org/content/10.110… 
The study only included people between 18 and 70 and excluded any individuals who had recently had COVID-19 symptoms, so the actual prevalence of immunity might have been a little higher even back then, but surely it was nowhere near the ~60% figure that everyone keeps repeating.
I have been really annoyed by people generalizing to the whole of Lombardy based on the results of studies done in just a handful of small towns for a while, especially since almost no details were available on the studies in question, so I was happy to find this paper.
I've looked at the literature on the prevalence of immunity for past pandemics and there is always a *lot* of heterogeneity. Even in the same country, the prevalence of immunity can be wildly different in different areas, so generalizing is a very bad idea.
In my opinion, the results of this study could be bad news (since it could mean that we're a *long* way from herd immunity even in the worst affected area), but it could also be good news (since it could mean that herd immunity starts kicking in much sooner than we think).
Of course, people will mostly opt for the pessimistic interpretation because they irrationally believe that we know the herd immunity threshold is *way* higher than that, but the truth is that we have no idea. I guess we're about to find out though.
ADDENDUM: To be clear, I'm not saying it's impossible that, in some small towns, the prevalence of immunity really was ~60% by the end of April. My point is that, *even if that's true*, it was silly to generalize to Lombardy as a whole from those data points, yet many people did.
I wrote "allegedly" because, to my knowledge, no paper has been published about the surveys in the small towns in question, so we have little details and can't even be sure those estimates are reliable. But this doesn't really matter.
In any case, if prevalence was less than 15% in Milan by the end of March in Milan (which is what this paper strongly suggests even if you make heroic assumptions), there is no way it was anywhere near 60% one month later, despite the lockdown that started on March 8 in Lombardy.
