Tagging tasks run by cloudwatch event


We recently started tagging all our resources with terraform in AWS to see in our billing section how are expenses distributed between our different environments. The last “issue” we have right now is that we can not see how to tag the task that runs a cloudwatch event.

We have a aws_ecs_task_definition with a tags block. Then we have a aws_cloudwatch_event_rule with tags also. And then a aws_cloudwatch_event_target joining the rule and the task, but (as far as I see) with no option for tagging.
When the task is run in AWS, I see no tags on that task.

Can anybody help me on this?

Kind regards,