No, you've been given a bum steer there, Sony or not .......
It may discharge if left on the PC for a long time (overnight is not recommended, for example).
However, it's perfectly OK to leave it overnight, or similar length of time, on the AC charger. You can't overcharge these batteries, and it can't "discharge" on an AC adaptor.
Your problem is typical of a battery that's shot, which it sounds like yours is !
You can try a soft reset, but I don't think it'll help.
Tell 'em the battery's failed, they should simply replace it - it sounds like it's still in warranty.
Anyway, the goods were/are faulty, so sling it back at them !
(You could also recommend they get the relevant staff to read up about things they "help" customers with ....... )
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