Evidence seems hard to come by on this subject, but a web search yields a lot of results along these lines:

https://www.quora.com/Which-uses-less-battery-for-a-car-audi...

It's hardly a scientific study, but it reinforces my prior belief that aux doesn't inherently require any electricity, while bluetooth is sending information over radio waves which of course has to use electricity.

I'm not sure about the difference when it comes to fancy, high-powered headphones, but my (anecdotal) experience with bluetooth vs. analog certainly supports the idea that bluetooth uses more power. I'd appreciate some citations if you have some other information about this subject.