If the panel is still functional and powered after unplugging the power supply, the battery is functional and connected.
Batteries lose capacity over time, depending on how often they are drained typically, so it may not hold a full charge. If you get a low battery trouble alert shortly after unplugging then the battery’s charge level is lower.
Just to verify, I see a lot of tamper alerts, are you unplugging the battery and plugging it back in or unplugging the power supply from the wall?