Default [Q] WiFi + Google Play Music issues

Hi guys,

Got a problem with my mother's Xperia SP. WiFi doesn't seem to stay on when the screen is off, despite the right option being ticked in settings. It doesn't show any sign that it's disconnected, but Google Play Music stops downloading the long queue of music we've selected to be stored on the device if the screen is closed for a little while. This is most frustrating, and I suspect a software bug.

This is an Xperia SP on Orange, build number 12.0.A.2.254 (the latest available, Android 4.1.2) and the bootloader is not unlockable or I would have installed Cyanogenmod already.

Any ideas?