I have an article right here that describes how to log into the Help Desk. It is much easier (and still trackable) to send an email to email@example.com, however.
As to your question, SQL versions are not on a 'per account' basis, so the upgrade would need to be done on the entire server. AS such, that is not something that is done on a customer request, but to be done during a future server maintenance.
I do see that the version on your server is 5.5.24 and the latest release from MySQL is 5.5.27. This is very close and if your database is from even the 5.5.27, it should be compatible as it is within the same version (5.5).
Our support team is happy to attempt this for you and let you know what any problems may be if unable to get it to import.
To do so, you would need to contact our support department (open 24/7) with that request. Be sure to validate your account with either the last 4 digits of the credit card on file, or the current Account Management Panel (AMP) password.
Additionally, be sure to upload the database file to your account and include the file name in the request so they can find it and perform the import.
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