And a quick google search even finds us some interesting laptop connector info.
From:
https://batteryuniversity.com/learn/...laptop_battery
The picture is a little deceiving. When I look at the actual connector, there are no electrical connections in the top and bottom big grooves. It must just be for alignment. The very thin grooves are the only ones with metal showing. There are 5 slots.
I am not sure if this works the same for these modem batteries, or if the pins do something else. I'll have to look more into the BMS info. However, it does seem pretty easy to just buy a male connector and get easy access to a 3S1P setup.