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While the card provider has the right to close the account etc. if in a credit balance, they generally don't.
The CSAs are reading from scripts - they have to tell you the "party line".
In practice, the reality is that someone/something needs to bring to the card providers notice that an account is significantly in credit to initiate such action.
There are several ways the card provider may become aware this. The easiest way is to tell them! :shock: It is also likely they use software to scan accounts regularly, but it seems they generally only act in exceptional circumstances triggered by various indicators (e.g. Amount of Credit balance multiplied by the period in credit could be one).
As I posted earlier, simply deposit the money and use.
I would go further and indicate it would probably be fine to put it into credit by not more than a small amount (up to $5K) and no more than a week or two before departure.
If you plan on using it for more, set up transfers as indicated by lovetravellingoz before you depart. That's a really good idea IMHO.