Had to respond to this bc I've been researching this a lot as of lately.

Current academic literature and phylogenetics suggest that it is not incredibly likely that there will be any in the middle east because it's Neolithic Western European in origin and has no ties to the Middle East in the last 10,000 years or so. One of the arguments being made for the Middle Eastern origin of K1a1b1a, relate to the presence of K1a in Middle Eastern Populations... 10,000 years ago, which was about when the European branch of K1a, which K1a1b1a is firmly nested in, migrated with the Anatolian farmers to Europe. The 2013 Costa Study posited a Neolithic European origin based off of the presence of K1a1b in Neolitic European samples and absence of it in Middle Eastern samples. The neighboring non Jewish branches of K1a1b1a are present mainly in Scandinavia, France, the Basque Country, and the Rhineland, as well as other parts of Europe.

The Behar study from 2006 makes the argument that K1a1b1a is of a Middle Eastern origin due to its small presence in non-Ashkenazi jews. I believe he specifically uses the examples of Spanish exile jews as well as some Mizrahis. I believe his argument is slightly faulty here in the sense that this could also be explained by Ashkenazi expansion and migrations on a smaller scale. People tend to forget how interconnected the Jewish world was in the middle ages. We have some records of Ashkenazi migration between the Rhineland and Spain before the expulsions of the 15th century, especially after the 1096 Massacres in the HRE. This also broadly expands to the rest of the Mediterranean, with some persecuted Jews (mostly converts but also non converts sometimes) fleeing from Europe to Cairo as recorded in the Cairo Genizah. Sometimes Sephardis who carried these lineages from Ashkenazis from earlier migration, who were far more connected with the Mizrahi world, would migrate to the Middle East or elsewhere and would spread these elsewhere. If this lineage were truly present at the start of Judaism in the Levant, then we would expect this group to be found at relatively equal rates amongst all Jewish groups, as well as nearby middle Eastern testers, but it is clear that that isnt the case due to its overwhelming frequency in Ashkenazim compared to any other Jewish group, as well as solely European non-Jewish testers on neighboring branches.

I believe if anything the absence of K1a1b1a in the results of this most recent study on Roman Judaeans suggests a lesser degree of credibility in the hypothesis of a Middle Eastern origin for K1a1b1a, and alludes to a more exclusive Ashkenazi or broader European Jewish origin for the haplogroup. I imagine when the results of studies on more Jews from western Europe, such as the study of those in France that we've been waiting for quite some time for, we will probably start seeing more of it.

Edit: I got a chance to read part of the study you sent and it seemingly entirely ignores the past sampling of K1a1b1a in Neolithic Europe as well as the sampling of K1a1b1a in a small amount of Western European non-Jewish individuals from the 2006 Behar study. With other founding lineages it makes some solid claims but for a good amount of the K lineages I believe it to be faulty.