You can see from the image exactly how wavelengths contract for objects in transverse motion in relation to a stationary source. You can even use the Pythagorean theorem and derive the formula for the Lorentz "contraction" from this image. Since the Lorentz "contraction" is exactly equal to Doppler shift, you can see that it's actually Doppler shift and NOT Lorentz "contraction" that causes the phases of separate beams of light to cancel in Michelson-Morley interferometers.

It's just Doppler shift.

It was always Doppler shift.

All you have to do is look at the picture. It's really, really obvious.

Relativity is just Doppler shift.