Well aren't you are a demanding one! 🤦♂️
I couldn't be bothered jumping through too many hoops for the likes of you, since I've already demonstrated the range of possible dates for the solstice, but a quick search shows the dates to be in the range I stated (note that a couple do extend into the very early morning of July 1, the latest being 1:07am, but those are back around the 8th century BCE). But it sufficiently demonstrates that for the period in question, the solstice was indeed consistently around 29 June (Julian calendar).
Jedael, S. (2016). Counting Days in Ancient Babylon: Eclipses, Omens, and Calendrics during the Old Babylonian Period (1750-1600 BCE). Unc Charlotte Electronic Theses And Dissertations. (Though the solstice dates are not central to the article's purpose.)