If the screen is LCD, screen burn should not be a problem. Electronic equipment tends to fail at switch on (and to a lesser degree switch off) rather than while working, so it may be better to leave it powered on with respect to reliability.
I have the CQ20 not the 12 do it has the mains power supply built in. I believe the CQ12 has a separate “brick” power supply and of course the switch on reliability is more applicable to that than the main unit.
One caveat is the ambient temperature, if that is likely to rise unduly while you are away from the studio, it may be best to power off. Of course leaving the unit powered on will use more energy.