The nearside front tyre of the 2006MY 4.2 SC I'm collecting tomorrow had a small cut in the sidewall and I asked the dealer to fit a new tyre. The current tyres are Toyo Proxes S/T 255/50 R20.

Back home I began to wonder if they'd be cheapskates and fit a cheaper tyre so I gave them a call to request same brand replacement. Salesman was studiously non-comital so I became more insistent and related a tale I heard somewhere about how even a minor difference in the diameter or rolling resistance of a different tyre can adversely affect the diffs on Range Rovers.

Is this true or just an old wives tale? 2006MY 4.2 SC
BMW R1250 GSA Rallye TE
Triumph Tiger 1050 (sold)
1979 Pontiac Trans Am (sold)

Personally, I would never mix brands on an axle for my peace of mind.
Also, It's a sign to me, that the car has been maintained on a tight budget...... Previously..
Vogue SE TD6
Defender 90 2.4
Defender 110 TD5
Vogue 3.5 EFI

Use the perhaps easy to research fact, that in Germany, you have to have both tyres on each axle, replaced, if only one is faulty. And both those tyres, have to be same.
It's not to keep tyre fitters busy.
Ask them, why they won't use an identical brand. . If it's time, then give them time. You are holding the trump card.... The money!

I always have all 4 matching

Different brands have different characteristics in different conditions - so by mixing, you upset the way the car was designed to handle. Accepted that a FFRR is not exactly a high performance car and 99.9% of the time it probably makes no difference whatsoever, but I'm always happier with matching tyres.

Wise words indeed. I was very uneasy at the prospect of having an odd tyre on one corner. Fortunately common sense prevailed and the dealer relented and fitted the correct tyre. All is well with the world.  2006MY 4.2 SC
BMW R1250 GSA Rallye TE
Triumph Tiger 1050 (sold)
1979 Pontiac Trans Am (sold)