[euromodder]

Using thinner gearbox oil is what just about all car manufacturers do these days, to get better FE ratings.

Volvo used lighter transmission oil on its later, more fuel efficient models that use the same gearbox, so I've done the same on my V50 when it was due for a transmission lube swap.
No immediate change in FE was observed - nor expected.
This isn't the kind of improvement that will show in day to day driving.