I read a sheet positing the torsen all gear differential has a frictional brake on the torsen bevel gears that allows them to lock to the spider gears - this is what they are concerned about having just the right amount of lubrication so the lockup is quick.
Apparently, limited slip additive really slows it down, and they are paranoid about synthetic base oil slipperiness However, their statement implies they did successfully test some of the synthetic gear oils without issue. I sense they don't want to be on the hook for any 'synthetic' base oil blend anyone comes up with.
I haven't had any trouble locking up with Mobil Delvac 1 gear oil. And we've had a bunch of snow to date - over 30 inches, and wind drift.
You might consider since mineral gear oil turns into molasses for us in winter.