I had the same issue with my 4 years old Xperia Z phone. For me only the right 1/3 part of the screen didn't work. And also it happened very sporadically. After some time the whole screen stopped responding but the volume keys and power button did work. Solution was to connect it to PC, after that the display was mysteriously fixed and I could at least reset the phone. But then after disconnecting from PC the screen stopped working again. Nice! It looks like just the fact of connecting/disconnecting the phone to the PC/charger affects this bug. Connecting it to PC again did help but then it went off again. Repair via Xperia Companion is finished - didn't help. My current guess is that it is related to the voltage i.e. that when you connect the phone to PC/charger the voltage goes up and it's enough to power the touch screen. But when you disconnect it, the battery doesn't have enough power to support it.
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