It looks to me like the extruder is going up immediately before/during the travel movements - that is ok and can be desirable, as long as it goes back down. Which would also jive with the fact that you say it prints the skirt fine, but when it goes to print the actual body of the object, the extruder has already moved up.
I’d start looking at your filament retraction settings,and in particular, the lift on travel settings. Not sure which slicing software you have, but in Slic3r, that would be found in the Printer Settings, then Extruder 1 menu. But that particular setting (Lift Z) doesn’t give different values for the ‘up’ movement versus the ‘down’ movement- it should be obvious that they should be the same. To diagnose, I would make that Z lift value something big - 10 mm, and make it a slow movement if you can. And then you could see if that is when the hotend is moving up but not coming back down. - which at least should tell you it’s a configuration issue and not a problem having to do with the geometry of your 3 towers, rod_radius or something else.
Is the printer ‘thinking’ it is already at a minimum Z, and therefore won’t tell the 3 motors to move down again after it has done the Z lift? What does your display tell you that the Z height is while it’s printing - both for the skirt, and once it has started printing in air?
Your blue sensor - is that a proximity sensor? Could there be some sort of crosstalk between cables or some other issue that is triggering it? I haven’t used one, but wonder if that is giving a problem.
Are you printing from SD card, or tethered through USB cable - what types of diagnostic/error messages are available to you while you’re printing?
On a different tangent, have you tried using a different slicer software to see if the problem goes away?