After trying to rearrange the prioritized Bibles, Logos started to switch those two Bibles on its own, blinking the Home Page section for the preferred Bibles and synchronizing all the time. Once I have ESV on the first place, in a moment my second Bible, and vice versa.
After restart of the program and even the computer it just does the same. I started with CTRL (logging) and my logo is attached.
EDIT: I am not able to attach log file. I will do it in the next post. Sorry for that.