Damage to differential?

[NDA]

Just having two new rear tyres fitted on my Range Rover - the fronts are fine.

The guy fitting them said that I should change all 4 tyres as any difference in tread depth would damage the differential.... I’ve not heard this one before.

[NDA]

The Range Rover is a tank - I can’t believe a few mil of tread depth would knacker the differential.

[NDA]

3mm of tread difference will 'destroy' a 4X4's coupling system?

Seems extreme.

[NDA]

It is a Range Rover supercharged.

I fitted 2 new rear tyres and so far it hasn't blown up.