No, that's not very possible. There is a small hole in the overflow valve that always allows fuel to excape from the injection pump. It is more likely that your WTS bulb has burnt out. Not that that is the issue, but it is more probable.
A friends truck is doing something similar but the wait to start light takes a few seconds to come on and the airdog does not cycle until the wait to start light comes on. Does the truck start fine? Does the airdog come on when you bump the starter/
exactly as long as the pump starts running when the starter engauges no issues. To prime the fuel system on my old 2000 the pump would not run with the key in just the on position, the starter had to be bumped for the ecm to send a signal to the pump.
well heres the thing....it used to come on when i turned the key and when i bumped the starter....just quit a few days ago.... i noticed that it had stopped giving a lil shot during the wait to start....then i put a fuel pressure gauge on 3 days ago and bumped the starter and cracked a fuel line to get the air out of the system and the pump never came on.....i ended up having to crank the motor with the fuel lines cracked goin into the block to get the air out.........truck runs fine and starts fine.......i just want to be sure its alright to drive like this.......i just changed the vp44 2 weeks ago when the factory lift pump died