i open a case and still Unassigned

I understand your concern about your unassigned limit increase case. This situation can be frustrating, especially when you're waiting to increase resources for your workloads.

Here are some steps you can take to help move your case forward:

1. Follow up on your existing case by logging into the AWS Support Center and adding a comment with additional information about the urgency and business impact of your request.

2. Ensure you've provided clear justification for why you need the quota increase, as this can help expedite the review process.

3. If your request is urgent, consider submitting a new service limit increase case from the Support Center with a higher severity level, and choose the Chat or Phone option for more immediate assistance.

4. If you have an AWS account representative or technical account manager, reach out to them directly as they may be able to help escalate your case.

5. In some cases, support tickets may experience internal routing failures. If your case remains unassigned for an extended period (several days), it might be worth opening a new support case specifically about the delay.

It's worth noting that quota increase requests don't typically receive priority support, and processing times can vary. However, if your case has been unassigned for more than a few days, this is unusual and warrants follow-up action.
Sources
instance limit increase EC2 | AWS re:Post
created a case for Limit Increase: Simple Storage Service (S3) still it is still unassigned. | AWS re:Post
Are support tickets typically unassigned after 11 days> | AWS re:Post