The centre of mass of the rod and the spinning disc drops a little.
Thus there is a loss of gravitational potential energy and a gain in the precessional kinetic energy.
You can think of it as the torque which is causing the precession rotating through an angle and so work is done by the torque which increases the rate of procession.
The change of angle of the rod is small and so the effect is difficult to see.
If you watch this Walter Lewin video at 2:30 he releases the gyro.
Does the inclination of the rod change?
At 2:40 a mass is added to the rod which increases the precessional kinetic energy and hence the rate of precession and there is a definite change in the angle of the rod.
The inclination of the rod seems to decrease when the mass is removed at 2:47 with the rate of precession decreasing.
The movement of the centre of mass down to ensure energy conservation also ensures angular momentum conservation.
The system starts with no precessional angular momentum in the vertical direction and then when released starts to precession with an angular momentum due to the precession in the vertical direction even though there are no external torques in that direction.
The precession angular momentum comes from the fact that when the centre of mass drops the component of spin angular momentum in the vertical direction changes (e.g. from having a horizontal spin angular momentum to the spin angular momentum pointing slightly down introduces a vertical component of spin angular momentum) and that change in spin angular momentum is exactly balanced by the increase in the precessional angular momentum so the net vertical component of the total angular momentum of the system stays the same.