I have yet to see that my external is charging my batteries. That being said I tried putting some fake batteries in the scope to see if it would start on strictly external power and it wouldn't.
I tried your experiment on mine and it won't start without good batteries installed. Perhaps is a difference in power packs but it would appear that perhaps like some other items on these scopes, what works on one doesn't necessarily work on all.
Also, my power pack is only supplying power to the battery box when the scope is turned on. I measured less than 1/2v when the scope is off.
Just sharing my experience:
I don't know if this will help us with answers, but... I have the LimeFuel battery. It has two 5.0 volt outputs. One is 1.0 amp, the other is 2.1 amps. With the batteries removed, the scope will not come on if I use the the 1.0 amp output. However, it does come on if I use the 2.1 amp output. Maybe it needs to be able to draw more than 1.0 amp for startup? That could be the difference in the various battery packs and what folks are experiencing?
Also, when powered up through the external source, my original AA battery box shows the amount of voltage grows the longer it is powered up.
It starts at 0.00 and grew to .51 volts in about 30 minutes. I don't know where it may top out. Would that charge rechargeable batteries over time? I don't know....but I lean toward the fact that it may serve to keep them charged. Once you turn the scope off, the external LimeFuel battery detects that there is no current draw and powers down. HOWEVER, he presence of that power in the box tells me I do not want regular batteries in the box with external power applied.
As concerns the sleep timer.... Mine
does not work if I only have the external battery connected. It
does work with only the internal batteries used. It
does work if both internal and external batteries are employed.
For me, at this point, I will be using the scope with both internal rechargeables and the external supply using the 2.1 amp input. I will set the sleep timer to 60 minutes so that the scope stays on and keeps the LimeFuel sensing a current draw. I'll push a button before the hour passes, keeping the unit ready to go. Maybe I can find another battery that has a 2 amp output, but doesn't shutdown if it senses no current draw. Then I could use the sleep timer.
Anyway, just my experience. May have value, maybe not...