If your precious Xperia ZR in standby mode doesn't turn on the screen immediately after your press the power button, don't panic. This happens to many people on all Xperia phones Z/ZL/ZR after the latest update to Android 4.3. If that is your case too, most likely it is not a hardware problem. It is a known bug on Xperia phones running official Sony Jelly Bean update.

In fact the phone works. The button press gets registered, but it takes some time to wake up. Maybe when you press it repeatedly, you turn it off again and then it seems it's not working at all. Or the first button press wakes up some process and only the second button press is registered as the real button press.

I suspect the delay is caused by something running in the background and in my case it was the stamina mode. It can also be something else. Maybe some other battery saving app or any other process running in the background. I have set up the stamina mode to keep most of my messaging apps running. When I turn off the stamina mode, tha lag is gone and the button works normally!