demonstrably
Not gonna happen. There are too many variables to control, and it's impossible to run the same project twice, let alone repeatedly. People with opposing beliefs will never be convinced by such a "demonstration"; they will simply find some detail or other to justify what they already believe.
The empirical literature on software projects is downright feeble. Meanwhile the programming culture is very rich in anecdotes and folklore. I find that suggestive.
Yeah, it's funny, I mean say McDonalds their whole business hinges on how quickly they can cook burgers, but no-one there would ever consider under-cooking as a way to speed things up. So the average software project manager is not even as smart as the average burger flipper.
Nah, you need an entire project that is demonstrably better executed than another and that attributes its success to unit testing (or whatever). This is where evangelists fall down. They shouldn't preach until they've gotten something tangible to show.