Personally I have already explained my view, but no genetic paper shows this minor influence (precisely from the Levant) in either Sicilians or Greek islanders, and as to specify, Greek islanders aren't so different at all from (some) mainland Greeks given the distance between southern Peloponnesians (with whom Sicilians plot, a result known since 2014 and confirmed by the peloponnese paper,
https://www.pnas.org/content/111/25/9211) and Greek islanders is small. so what Greek islanders have likely also Greek mainlanders have.
It is far, far more likely that the Levant-like admixture in Sicilians in those models is north african (pre islamic-trade SSA gene flow), and actually I do not trust the samples you find in online calculators, especially when it comes to certain peoples that have been "trolled" relentlessly during the years (still now go take a look at the "sicilian" and "south Italian" results at the 23andMe subreddit from accounts who only post about south Italians and greek Islanders and talk about their having 50% (sic) "WANA" admixture). Naturally I hold my skepticism because of that trend and because of the discrepancy between online results and those of academic studies ( not that every study is the Truth, but usually they are more trustworthy and you can still say what they got wrong because of a certain transparency, whereas in the calculators I've seen online I can't get to know whether the samples are genuine or not.
According to the literature, south Italians don't have any levantine admixture and Sicilians and some Calabrese have some north african admixture, while as for Greek islanders I haven't seen many studies but judging from the results of the Cretan paper they don't seem to harbor any such ancestry (at least it wasn't detected, I recall, but if I am mistaken feel free to correct me of course).
The only paper I recall that supports this view is the "east med continuum" paper by Sarno, but it had serious flaws, as you can check if you check the "near east" label that spans populations from Anatolia to Jordan, so I think that the "near east" part in the admixtures was anatolian-like rather than Levantine-like.