I don't know, but I presume. In my runs Lombardy and Tuscany have both increased levels of Deu_MA-like ancestry. Its also interesting that in this recent study on Italian yDNA the samples from Bergamo valley and Volterra have both increased levels of R1b-U106. And I-M253 is present throughout Italy in non-negligible levels too.
Also there is a Slavic-like component present in my runs, this in regions like Apulia and Calabria too. This is another source of Northern shifted ancestry and its related to increased levels of R1a, compare again with the study and the increased levels of R1a from a Slavic-like or Germanic population, also in places like Borbera valley. Of course, we don't know when exactly it came in to Italy, but especially for R1a-M458 its more likely a later influx spread it imho and this in turn would make the spread of R1a in general more likely related to a later migration.
Recent yDNA study:
https://pubmed.ncbi.nlm.nih.gov/29382284/
Quick check for M458 and some clades of Italians:
https://www.yfull.com/tree/R-M458/
Apulia and Calabria pick up additional Slavic in my runs, which aligns well with the yDNA.
I might quote on that from the historical perspective:
From: The Age of Robert Guiscard: Southern Italy and the Northern Conquest. In Norman times additional North and West European people came to Southern Italy, but like we see in this example, also Slavs from the Eastern side of the Adriatic. These left a mark on the land, they surely did. It also explains the sometimes significant differences between different subregions and even villages in the South: Both ancient and recent settlement patterns could vary quite a lot.
Its worth to mention that Sicilians picked in my runs nothing or close to Nothing from the Early Slav, but regions like Apulia, Basilicata and Calabria get the highest hits because of both ancient and very recent migration from the Eastern Adriatic region, not just Illyrians and Greeks, but also Slavs and Albanians. How much is ancient, how much is recent, we will see. But just check some R1a-M458 from Italy. Even some of those which made it to Sardinia, and the resolution for the Sardinian samples might not always be ideal, has a TMRCA of 350 AD (!) and is closest to Scandinavians and Slavs. Another sample same age 500 AD, clade with Slavs.
So its not like there was no migration. We have autosomal affinities (percentage of DEU_MA or Czech_Early_Slav respectively), we have yDNA (I1, at least R1b-U106, but also other subclades of R1b and E-V13 in all likelihood, as well as R1a-clades like M458, and we have historical accounts of Lombard and Gothic settlements, the Lombard ones attracted additional Frankish immigration, you can check that in the study on Collegno, and we have Slavic and Albanian immigration, also changing the landscape.
Its quite striking that the Northern shift in Sicily got more Germanic-like and Apulia more Slavic-like in my runs. And looking at the history and uniparentals, I don't think that's just a coincidence and rather shows what can be picked up. From Grugnietal et al. especially figures 2 and 3 are quite informative. For example the variation of clades of R1b in Italy. There are peaks of relative percentages for U106 in the North and Tuscany, but also in Sicily (!). Apulia sticks out with R1b-M412, from p. 48:
They seem to think most of the R1a is ancient, but they can't be sure, especially since M458 is present:
M458 is at least present at a level of 1 percent. But more needs to be done to be sure about its origin.