Alternative power supply for Stick Up Cam Battery HD 3rd Gen

I have a Stick Up Cam Battery HD 3rd Gen. According to this article, “… indoor/outdoor power adapter (sold separately) … for nonstop power”.

I believe that adapter is this one: Indoor/Outdoor Power Adapter (Barrel Plug). I can see its output is 5V DC, 2.5A .

What are the details of the barrel socket, such as inner/outer diameter and polarity?

Any help would be much appreciated.

I was provided by Ring the same camera to replace my first generation Stick Up Cam which was battery operated and if plugged in, could run on AC power (it was always charging). I can’t get the Stick Up cam to turn on with the barrel adapter if there’s no battery inside and when there’s a battery, it does not recognize the barrel adapter which is plugged into the wall. How are we to use this barrel adapter?