Just noticed that the "LastVersion" file in the profile folder, as the name suggests, gets updated when the profile is opened with (another version of) BAS e.g. from 114.0.5735.91 to 117.0.5938.63. Therefore it's not suitable for our need to determine the original version of which a profile was created.
But at least can be said what was the latest BAS version that worked with it ;)
How to automate Google Chrome's profile?
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You can set extension like this https://chrome.google.com/webstore/detail/get-cookiestxt/bgaddhkoddajcdgocldbbfleckgcbcid
in yours chrome and then import it to the BAS with module like this
https://community.bablosoft.com/topic/18824/cookiestools/62
if you have any problems write to me in pm.