Does anyone have any experience of how long CX takes to process refunds?

I book a refundable flight for just over $5k and had to cancel. I went through the rather elaborate refund procedure and was told it would take 7 business days. It is now the 11th business day and still no refund showing and now my CC is due. Trying to get hold of anyone at CX who can assist is proving challenging so I am tempted to just dispute the original transaction on my CC and let CX deal with Amex instead. Thoughts?