I tried both the native oscilloscope probes from the kit and from the Rigol DS1054Z kit (this oscillator has slightly better probes both in frequency-amplitude and phase characteristics).
All probes are connected in parallel to the generator output, ground probes are connected at one point of the generator ground.
Something else I noticed:
1) First 4 photos: Default setting (Ch-Ch Skew = 0.00s) in channel 1 settings. The difference in phase shift between 1 and 2, 1 and 3, 1 and 4 channels is about 20 degrees, but if you turn on three or four channels at the same time, the shift is eliminated.
2) Last 4 photos: When setting (Ch-Ch Skew = 750ps) in the settings of channel 1. The phase shift difference between 1 and 2, 1 and 3, 1 and 4 channels is about 1 degree, but if you turn on three or four channels at the same time, the phase shift becomes about 20 degrees.
Why this happens is unclear.