What you've linked has essentially nothing to do with the discussion. What the person who replied to you thought you were talking about was the equation for energy stored in a capacitor, which is proportional to inverse of distance.
I'm not a physicist, but my understanding is that batteries do not use the electric field at all to store charge, unlike a capacitor. Batteries use chemical reactions to move electrons, not static electric charge.
It's linear, and it only applies to capacitors. Batteries are a completely different beast.
The cost reduction we have been seeing for batteries has completely different reasons, and no relation at all with that.