I find it highly unlikely an automatic would have better fuel economy than a manual.
I saw a study from, I think it was Volvo, that basically concluded that while trained drivers focusing on driving efficiently could get better fuel economy out of a manual, 'normal' people driving normally got better fuel efficiency out of an automatic. And this was a good 10 years ago, so automatics have only gotten better.
Training on manual cars also improved last 10 years. Don't know about other countries but here fuel economy is a mandatory part of training. Recent manual cars provide indicators on when to switch. You can combine that signal and training for better fuel economy. TBH, I don't know how that compares with automatic.
I saw a study from, I think it was Volvo, that basically concluded that while trained drivers focusing on driving efficiently could get better fuel economy out of a manual, 'normal' people driving normally got better fuel efficiency out of an automatic. And this was a good 10 years ago, so automatics have only gotten better.