On Xperia 10 III the finger-print scanner can wake the device. I.e. when you put your finger on the power button but don’t press it, this will wake up the device. However, it will still be locked. To unlock it, you have to remove your finger from the button/scanner and scan again.
Please make it so that the first finger-print scan that wakes the device also unlocks it at the same time. This would make it much quicker to start using the phone.