HP Elitebook laptop fails to switch from mains to battery when AC power is lost

I have acquired two (identical) used HP laptop power supplies compatible with and for two identical HP branded laptops in an office. When power is cut to the supply, the attached computer immediately shuts down cold.

I imagine the power supplies must be faulty for this to be the case, but both supplies are exactly the same in this behaviour, perhaps there is a voltage spike that the laptop is protecting itself from.

Is this a common fault? Could it be water damage? Do some supplies just do that and it's annoying but normal?

Update:

Laptop model: HP Elitebook 8540w

Firmware initially not updated, after solution: F.60 Rev.A

Solution did not work until the battery was sufficiently discharged.

Update 2:

With battery fully charged, solution reverts and issue re-emerges as before.

Update 3: Additional information

The problem was compounded with the installation of a new battery. The issue arises when the connected charger is disconnected from mains power. The computer with no warning immediately shuts down. It does not occur when the charger is disconnected from the laptop first.

The problem is unlikely to be OS related as it can be forced to occur even when the the boot menu is displayed.

More specifically the shutdown occurs when the battery is full, if it is charging from part-full, the problem does not occur.

When the computer is in hibernate mode, the issue does not occur, when it is fully powered, it does, even if only up to the boot screen.

This is a high-priority issue as I work in places where mains power is intermittent, for periods longer than when the battery itself can sustain the device.

The first solution that appears to have worked for one of these two identical computers is a BIOS update. That machine no longer has the fault. The second machine retains the fault even after upgrade.

The first machine had its BIOS upgrade done via USB stick, the second used the HP Windows utility.

Hibernate or On | Low/high voltage pulled first | Battery at partial/full charge : Fault?

Hib | LV | P : No

Hib | LV | F : No

Hib | HV | P : No

Hib | HV | F : No

On | LV | P : No

On | LV | F : No

On | HV | P : No

On | HV | F : Yes

Answer

As it turns out, the replacement batteries are the root cause.

Replacement batteries from two more reputable manufacturers both work just fine. Incidentally the problem batteries marked as 5200 mAh are in fact 4400 mAh by Samsung when viewed through software, so falsely marketed and illegitimate. The batteries at this link I would not recommend.

This and this battery both work and have the capacity advertised when viewed in software.