How long have you had their accounts. As of late, with talking with their Social Media team and Executive office, they've been pretty stellar at addressing my concerns. I closed one card and prior to closing the second, I hit up the ExO. Needless to say I've gotten everything I asked for... no AF, better card, and a CLI. I think they're starting to realize they're bleeding customers and their abysmal CS needs to go.
I had the same situation early this year. I had my basic rebuilder platinum since 2008 and I nearly closed it back in March since it seems like they could care less. But I kept it open although it had an AF of $39 with a $1100 CL.
Fast forward, I contacted the EXec office and they bumped me up to $5K and PC'ed me to the Cash card.
Not to say that it works all the time but sometimes it pays to wait. How long have you had the card? I showed them that i had longetivity and a solid payment history with them. And they even removed my AF. So sometimes at was at the end of my wits wih them so i know the feeling 
And how did you approach this CLI? Chat? Phone? Again how long you have had your accounts?
You called the ExO cold? Did you send an email prior and did someone reach out to you? They'll treat you like crap when you don't have a case opened for them to look at. But young is right, if you want to close them, close 'em. I was in the same boat, they've impressed me enough the past week that I'm giving them a second chance. In away, they gave me a second chance at credit, so it's full circle.
