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Yes, this is correct. I find it very strange that he does not mark the 5000mAh as "fake". What exactly does he mean by fake anyways? Fake as in "not holding the advertised charge" or fake as in "is not authentic based on the brand"? If a battery advertises as being able to hold 5000mAh and then tests shows that it can't, I would consider that battery a fake. And as a battery of the size of a 18650 just physically CAN'T have more than 3500mAh, ALL batteries advertising more than 3500mAh should automatically be considered fake.
3.2v? That is not correct at all. Just look at the datasheet of a LG Choco 18650 battery. It's cut-off voltage is 2.5v. Even most battery chargers like the IMAX B6 will discharge down to 3.0v when testing batteries.
Why do you need the Raspberry Pi? Can't you just implement a HomeKit Server on the ESP8266 directly?