I think from a purely mechanical point of view, this is the case, meaning that if you have two cars, one with manual gearbox and one with an automatic gearbox, both driving at the same speed with the same gear ratio, the one with the automatic gearbox will likely use slightly more fuel. But even this might not the case at all possible speeds. But at some speeds when the car with the manual gearbox is using the wrong gear, it might use much more gas. The problem with the manual gearbox is not in the gearbox but the human using it. First of all, not all drivers know when is the best moment to change gears and even if they know, they are not always able to change gears at the right moment. When you are turning on a crossing and accelerating, switching gears can be difficult.

Also automatic gearboxes can change gears usually much faster than humans. Operating the clutch is hard and when not done in the correct way, letting the clutch slip too much, waste energy.

Taking this all into account, in many cases, an automatic transmission might be more efficient.

I have been driving cars with manual transmissions all my life. My current car as indicator lights telling you when to change gears up and down. I often find myself skipping gears, especially when accelerating in the city, I often skip fourth gear. The car has six gears and you should already switch to the fifth gear when driving 50Km/h. My first car only had four gears, and the fourth gear was only used when driving over sixty. If a car has more gears it also means that you have to shift more often, which might make driving less efficient in city traffic due to the many more moments you shift slightly at the wrong moment.