The relevant portion of the PRO-32 schematic is attached. The battery is connected to the power jacks through inductors L110 and L111. If either is an open circuit, the connection from the battery is broken. It is not likely that the inductors are open circuit. R195 is a 39 Ohm resistor used to control charging current. When powering the radio from the battery, the switch in the J7 PWR/CHG jack must be closed, shorting the 39 Ohm resistor. Plugging the AC adapter into the PWR/CHG jack puts the 39 Ohm resistor back into the circuit by opening the switch on the jack. The J6 PWR jack also has a switch inside that can interrupt the connection to the battery. To allow the battery to power the radio both power jacks must be unplugged and both internal jack switches must be closed.
If you remove the battery pack and unplug the ac adapter from the wall socket the battery circuit can be checked with a meter that measures resistance. Measure across the pins of each power jack and see if both switches are on. To easily check J7 just measure across R195 and see if is zero Ohms. Find similar convenient points to measure J6.
If one or both jacks has a faulty switch, spray contact cleaner (Deoxit preferably) into the jacks. Place the radio on its side with jacks facing up to let the cleaner work into the contacts. The plug an unpowered AC adapter into each jack about a dozen times to exercise the switch in the jack. Do the resistance measurements again to see if the switches make contact now. If not rinse and repeat with the contact cleaner.