@max I measure 4.5v DC at 1 & 2 with my charger connected. From connections 3 & 4 I measured 4.0v DC with the charger connected. With the charger removed, 3&4 measured 3.8v.
The charging cable is a simple straight thru USB-A cable. From the flat end with the pins facing the floor, pin 1 on the left is your +5vDC, pin 4 on the right is your ground.
check the continuity of your cable; If nothing start at the round end. The magnetic connector seems rather cheap. The magnetic connections make contact with the wires via the pressure of the plastic housing they're pressed into. Use needle nose pliers to pull the magnetic bits straight out and test continuity again.
If the battery leads are shorted, the charging LED does not come on. If the battery leads are disconnected, the charging LED flickers faintly.
With the battery disconnected and the charging leads connected in their regular position, I can power the unit on.
If there's anything else you'd like me to test or measure let me know, I'm happy to help..