ECS Fargate Task Losing Internet Connectivity

Running a task using aws ecs run-task works well for one-off tasks, except for they always lose internet connectivity within about 5 minutes from starting.

Here is the script I'm using to start the task:

export AWS_ACCESS_KEY_ID=${AWS_ACCESS_KEY_ID}
export AWS_SECRET_ACCESS_KEY=${AWS_SECRET_ACCESS_KEY}

cmd=$(cat <<EOF
{
    "containerOverrides": [
        {
            "name": "${CONTAINER}",
            "command": ${COMMAND}
        }
    ]
}
EOF
)

taskDefinition=$( \
    aws ecs list-task-definitions --region us-east-1 \
        --family-prefix ${TASK_DEFINITION} \
        --output text \
    | grep ${TASK_DEFINITION} | tail -n1 | awk -F '/' '{print $NF}' \
)

echo "Running '${SCRIPT_NAME}' using '$taskDefinition' task definition"
aws ecs run-task --region us-east-1 \
    --cluster ${ECS_CLUSTER} \
    --launch-type FARGATE \
    --started-by "ManualScript" \
    --task-definition $taskDefinition \
    --network-configuration "awsvpcConfiguration={subnets=[${TASK_SUBNET}],securityGroups=[${TASK_SECURITY_GROUP}],assignPublicIp=ENABLED}" \
    --overrides "$cmd"

The container name and task definition family do not have auto-scaling rules, which was my first thought, so I'm not sure how to identify the fix here.