It may well work providing the voltage doesn't sag but I wouldn't be trusting it because, the back light will be bigger in size and may require high current and voltage to run them at the correct colour temperature and since a larger screen has more pixels this will require more chips to drive them which will result in more current consumption of a kind.
The power supply may well look the same but may well have component variations in the Primary and secondary sides. I reckon the clue is 'one size doesn't fit all' so to speak otherwise they wouldn't categorize them for sizes.
Dave