Leadscrew whip is the least of your worries.
When the spindle is moving along the gantry, and stops, then reverses direction, your entire gantry is going to oscillate back and forth like a pendulum, probably moving about 1/4", if not more.
Unfortunately, you've based your design on the least rigid machine out there in this size range.
Your gantry sides, motor mount plates and screw support plates need to be made much stronger, or you won't be able to take advantage of the speed that the 5 start screws provide.
The gantry sides should be a minimum or 2 layers of plywood laminated together. Even that will flex. A torsion box like side would be much stiffer, but will be considerable larger.
Also, a triangular shape would be much stronger than the "L" shape, as there would be a lot more material to reduce flexing.