Physics dictates that pitch is a function of string length, string mass and string tension. Height has nothing to do with it. If the sounding string length and string mass remain the same, then the tension required to maintain a base pitch of say 55 Hz on the A string remains the same. Lowering the height just narrows the distance one has to press the strings, which costs less physical effort.
The further one increases the string height, the further one is stretching the string in order to bring it into contact with the fret, hence the more under tension from the base value the string becomes when fretting compared to lower string heights.