That doesn't seem right.
We have had a Google Nest Battery Camera since the first week of September 2021, plugged in all the time with the optional 10-meter weatherproof power cable. It has to be plugged in with the cable feeding to the FRONT of the camera (as odd as that seems). Our camera always shows as "Live" with a livestream video when we check it in the Google Home app. If we check Settings | Battery while it is plugged in, it says "Plugged in" and shows an infinity symbol. Only if we turn off the power to that circuit and go to Settings | Battery can we see the battery's charge level, and it always shows 100%, even on cold Minnesota days and nights as low as 17 degree below zero Fahrenheit.
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