No. The observer is at rest in a gravitational field. The horizon is at a fixed distance below him. (To see this, at any instance of the observers proper time, you can define a momentarily comoving inertial frame (in which the observer is stationary) which is just your usual Minkowski spacetime. The distance from your observer to the horizon is surely constant.
To see that the distance to the horizon is not
constant, look at it from the gantryís frame (the frame of someone floating in the flat spacetime in which the observer is accelerating). The observer always moves at < c in that frame, whereas the light (communication from someone below) approaching the observer moves at c. Then the gap between the beam front of the light and the observer is ever narrowing. (It narrows ever more slowly as the observer approaches c in that frame, such that the beam front never reaches him.) I donít see how your argument using an MCIF shows otherwise. The effect of this narrowing gap is that, the longer the observer waits to stop accelerating, the sooner the chasing communication will then wash over him (by his clock).