No, I'm afraid not.

Assuming that the aircraft is in a vertical dive with the engine at idle it will speed up (accelerate) until the force of gravity equals the drag force. From that time on the aircraft will fall towards earth with constant velocity, i.e. with zero acceleration. When this happens fuel will accumulate at the front part of the tank due to the force of gravity. Remember that the fuel itself is in a sealed compartment and not subject to air resistance.
As an analogy consider yourself in an elevator going downward at constant speed. Pour some water out of a bottle and observe where it will end up. Presumably on the floor. This is true for any speed of the elevator as long as it is constant.

Interestingly enough the fuel will accumulate in the front of the tank even during the first phase of the vertical dive, when the speed is increasing. This is because air drag causes the acceleration of the aircraft to be LESS than the earths gravity, i.e. 1g (32 ft. / second squared).
The fuel in the tank however is free to move about at will relative to the surrounding aircraft and since it is not subject to drag forces from the air it will fall towards earth with an acceleration equal to 1g. The front of the tank will of course stop the fuel from falling out of the plane :-) and hence this is where it will end up.

The only way to get the fuel to collect in the rear of the tank during a vertical dive is if the thrust of the engine is maintained so that it is greater than the drag forces acting on the aircraft.

Since the stiffness of the fuel tubing prevents the clunk from falling forward towards the front of the tank, the reason the engine doesn't die during the dive is that it survives on the fuel available in the fuel tubing between the clunk and carburettor.

IMHO, the main reason that the engine doesn't stop during the dive is that the dive isn't maintained long enough for the fuel in the fuel tubing to be used up.

/Red B.