Answer
No, it will not precess.
Work Step by Step
Under the circumstance described, there will be no net torque on the gyroscope. Gravity pulls down on it at the center of mass, along a line that passes right through the pivot. The pivot provides an upward force of equal magnitude, upward along the same line. There is no net force and no net torque, so the gyroscope’s linear momentum and angular momentum are conserved, and no precession takes place.
