The mystery has been solved! It was clear that the problem was around the anode planes, so I started changing the components around that part, thinking that maybe something is defective. I first changed the anode driving shift register, that did not change anything. Than I changed the MOSFETS of planes 2 and 3 and... surprise! Now everything works well. They must have become damaged in the first phase when I had some wrong connections... I truly hope that they did not arrive damaged form Farnell, because 2 bad out of 4 would be very concerning.