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Well I think Chase was telling you
you want the CSR. But they only pre qual'd you for the CSP. So wait until your HP's age and then see if there is a pre qual for what you really want.
I know they do PC but I think you have to wait 1 year. So you might as well get the card you want right from the start.
With Chase you can get "pre-qualified" or "pre-approved"... these are 2 different things. "Pre-qualified" is just an offer to apply for a card, this is similar to getting those offers in the mail. You have about the same chance of being approved as if you cold apped for the card. If you see "pre-approved" in the Just For You offers on the website, and the details show a specific CL and APR% (as opposed to a range of APRs), then you're pre-approved and it's (almost) a sure bet you'll get the card if you apply. Pre-approved offers can also bypass 5/24 in most cases, while pre-qualifications can still be denied for 5/24.
But in any case, I'd wait until your youngest account is over 6 months before attempting to apply. Longer is better, and if you can garden long enough to get most of those inquiries off your report that will improve your chances as well.
Sometimes you can get in-branch pre-approvals that don't show on the website. If there is a branch near you, give that a try when you're ready to go for a new card.
If you find you can't get CSR, get the CSP (it offers 60K UR points for $4k spend, compared to CSR's 50k points). Keep it for a year, and if its limit is 10k or higher (or you can get a CLI to this amount), you can then upgrade it to the CSR.