Barclaycard Security Mess: how long until they close the card?

My Barclaycard was locked due to "unable to verify address change". After a lot of back and forth I've become fed up, and I'm done doing business with Barclays. I closed my savings accounts with them, and I'm giving up validating the address change with them.

I was able to call and pay off the card, and as far as I know, the balance is $0. The last person I talked to refused to tell me the balance, despite the fact that I've had other agents who will tell me the balance, and let me pay the card.

Despite the lock I'm able to see the balance in my financial aggregator website, and it shows as $0. I think I'm just going to abandon the card, as it's locked and no more charges can be made to it anyway.

So my question is: how long until Barclaycard closes the account on their own?

Finally: I would strongly recommend avoiding Barclay and Barclaycards.