If you are making a list of pros and cons you (at least so far) don't have anything in the "pro" column for closing it. You said you are maxed out. If you have the money to PIF now then do it before you close the account. If you close your account and continue to pay the balance off in a few months, then for those few months the scoring will treat your card exactly the same as if it was open. It will count the balance and the credit limit into your utilization.
Simplified, closing with zero balance is good, closing with a balance is at best neutral at worse it looks bad on a manual review.
I will toss one thing into the "pro" column for you. It possible you are being "balance chased". Meaning they plan to continue to lower your limit as you pay it down. (Search the site or Google for more info) In that scenario closing the account would help because as you pay it down scoring will use whatever the limit was when you closed the account as the number to calculate utilization. This would help your scoring vs a liimit that continues to decrease monthly.
I know you are pissed at them and want nothing to do with them but your credit is more than your score. Banks look at patterns to trigger action like this. Example are you balances on your other cards escalating (both other Synch cards and other bank cards), new accounts, new inquiries, is your profile overall strong or weak etc. etc.
First of all I would contact Credit Solutions 877-379-8175 and see what the deal is with the balance chasing.
Secondly Sync only HP's on new accounts with exception of 1 or 2 cards they manage.
Third..... sometimes there's an auto CLD when a card isn't used for X months before they actually just close it for non-use. My example would be 22 months in which they closed an unused account.
Closing the account will freeze the CL at it's current amount of 610, you make payments under the current APR/Promo rates as required and the UTIL will decrease as you pay it off.
To look at the bigger picture of your profile you will need to post more info regarding all cards/limits/use Current FICO scores if you have them and any apps/approvals in the last 12-24 months.
As stated, don't close it until you pay it off.
A bigger problem is why they balance chase you. You don't want that to spread. Reputable lenders don't do that without a cause. What is your entire credit profile look like? Score? How many loan, cards? What kind of balance you are carrying?