Kossel accumulates error along Z axis
After some more testing I checked that the problem shows up only on Z tower and is not related with current. Examining the tower I found very dumb mistake. Pulley on the motor didn't have a setscrew! Don't know how could I missed that. The fact that it could somehow run is even more amazing. Thanks @tom-van-der-zanden and @darthpixel for help. It is for sure that such synchronization failure could have been caused by current shortage as well.

UPDATE:

Oh! Y tower pulley doesn't have setscrew as well! But it was running smoothly and in sync with X (which has that screw). Amazing. - UPDATE2. Not really. After closer look only Z tower misses the screw, which explains all.

UPDATE3:

After setting a srew into pulley the problem is completely gone!)