The "3-G" rule (vaccinated/recovered/tested) no longer applies to enter public places, universities and restaurants. Although it still applies to air travel and to long-distance trains and busses.
Wearing the FFP2 face mask in public transports, essential shops, restaurants, public transport and in the conference hall may still be mandatory in June.
To be updated on the rules in force in Italy and the regulation to enter the country, please consult the web-site of the Italian Republic Ministry of Health:
https://www.salute.gov.it/portale/nuovocoronavirus/homeNuovoCoronavirus.jsp?lingua=english