so what is the full service tag data?
3710b or 6710b, you posted both , and is not complete at all those numbers, (vast submodels exist)
the former number dont fly..
if the PC run ok, on AC pack (HP spec, pack) no battery installed then it (pack) and the PC are ok (no bad jacks)
leaving 2 things, battery (that love to fail) or the charging chip you can not unsolder, ever and replace.
the center pin, seen on the schematic
or is the AC line loss early warning. (tells PC to go to battery early)
this signal ends up on the charge chip
called Charge Control line.
the schematic is free.
have no battery pack schematic that drives this pin
but its tells the PC to go to (battery early, as power fails AC)
the changer over is not instant
so they this pin tells the PC early, the power is failing
my guess, it can detect power loss, in the pack, in 2 milliseconds flat. (power loss early warning)