I agree that the body diode will no longer carry current at forward voltages below the bandgap.
However, this FET gate don't contribute to the cutoff mechanism.
When it is off, it's just acting as a diode.
It can be said that the cutoff depends entirely on the body diode.
See past topics below for more information.
https://forum.arduino.cc/t/fdn340p-drawn-incorrectly-on-schematic/145037