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I don't think HPs are worth a CLI unless they significantly raise your CL. Personally, I wouldn't have the majority of my cards with only one lender. I consider this risky. A CSP requires a $5K minimum CL, and right now your Amex is your largest limit. Chase doesn't like to be your first BFF with a large credit line. I would let the Anex age to at least a year - so it shows you can manage your limits - and see if you have a pre-approval for the CSP.
ditto