I built two gCore prototypes and now have a few weeks experience with them.  The dual battery system seems to be working fine on one unit.  The batteries remain at the same balanced charge level for long periods of time when the device is off.  On the other unit the batteries would discharge over a few day period.

Experimenting

I did some experiments with the failing unit to see if I could understand what was going on.  Unfortunately the results were inconclusive. 

Charging until the charger terminated the charge cycle would result in both batteries being charged to within 10 mV of each other.  Based on my original internet research (yeah, I know, don't believe everything you read on the internet, even if it sounds good...) this should be sufficient for a pair of balanced batteries.

I attached voltmeters to both batteries and put an ammeter in series with one battery with gCore turned off.  I saw a small current flowing between the batteries, usually between 1 and 3 mA.  At first I thought that this was just the batteries finding equilibrium and that it should stop after while.  But it didn't.  And the battery voltages ran down over a few days.

I then shorted both PTCs so the batteries were directly in series and repeated the experiment.  Same results.

One battery always had a slightly higher voltage, even as they discharged together, and there was always a small current flowing between them.

I can understand the battery with the lower voltage drawing some current from the battery with the higher voltage but I don't understand why its voltage declined over time as well. 

I'm pretty sure there were no other current draws on the board.  I verified this by removing one battery and verifying the current draw was essentially zero.

The batteries I used include their own protection circuitry.  I don't know if that somehow interferes with the balancing.  The other gCore prototype hasn't shown this problem and when I measure the current between batteries while gCore is off I only see ~20-30 uA.

So I would say my attempt to use the ICR10440 batteries in parallel to maximize use of the space was not a success, although it does seem to work some of the time.

Back to a single battery

It's easy to use a single battery with gCore.  Remove (or don't install) the AAA battery holders and connect it to one set of terminals.  Jumper that set of terminals PTC.

I attached an old iPod Touch battery with 3M double sticky foam tape and gCore works fine.  Best probably to use a 3.7v LiPo or Li-Ion battery with between 600-2500 mA capacity to match the charge current.