The argument at least I make is that there was some gene flow from Roman slavery and it is not specific only to the East Mediterranean, though to answer your question or statement that you mentioned with West Asian and Levantine slaves were only specifically brought to Southern Italy as an argument by some on other forums, I don't think it was only restricted to this area, there is 2 reasons for me one is logical and the other is an example that it could have happened and why Southern Italy. At the time Southern Italy was still vastly Greek speaking, it would make sense for slave owners of the South to get Greek speaking slaves and the East Med, Anatolia, Egypt and Greece/Aegean were Greek speaking in the sense it was like a lingua franca. The second point which I even made on Anthrogenica is take the example of the first Serville War, the leader of the Slave revolt was Eunus a Syrian slave and his partner was Cleon a slave from Cilicia, so we know that Eastern Mediterranean slaves were present at least during this period, how much? We cannot know. Was it only restricted to Southern Italy, definitely not.
Under this scenario, I would expect Rome and nearby urban areas of Central Italy to be among the most affected (sheer demand) parts of Italy but that doesn't seem the case. And in term of proximity/logistics wouldn't North Africa make much more sense for Southern Italy rather than East Med?