Can EKS Fargate be used in a private EKS cluster which has no outbound internet access?

According to the AWS documentation, the aws-alb-ingress controller is not supported for private EKS clusters with no outbound internet access:

https://docs.aws.amazon.com/eks/latest/userguide/private-clusters.html

AWS Fargate is supported with private clusters. You must include the STS VPC endpoint. For more information, see VPC endpoints for private clusters. You must use a third-party ingress controller with AWS Fargate because the ALB Ingress Controller on Amazon EKS does not work in private clusters and because Classic Load Balancers and Network Load Balancers are not supported on pods running on Fargate.

Unfortunately AWS provides no suggestions here on what the third-party options would be. I have not been able to find any information specific to EKS Fargate Private Clusters.

Questions:

1.) Is there an open source ingress controller that uses ALB that would work for Fargate?

2.) Is there a specific reason why the aws-alb-ingress controller will not work in a private cluster? I might be able to request outbound access for specific ports, if that is the issue, but AWS does not provide any detail on this.