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Anyway, I think that once you loose traction because of differential, there is no tyre that helps. In my case, even with the best tyres, all I would have moved is air.


That's probably true for the passenger rear in your pic because you have a floating tire. But both front tires seem to be firmly planted, and the extra dig that good tires provide can prevent an open diff from sending all the power to the weak side because aggressive tires are less likely to lose traction.
What make/ model tire do you have right now?

97 Montero SR: Factory rear locker, 3.5 L 24 Valve V6, ActivTrac 4wd (Love it) , 3 way suspension, four wheel disk brakes, 32x11.50 Pirelli Scorpion ATs