Influenced Opens is not an available metric if for an automated push -- a push notification that has a behavioral trigger on it. This is a feature is not yet available. However, the following workaround can still meet your needs:
Since automated pushed track event conversions, a custom event can be added to track app opens, among other actions, that happen as a result of the push being tapped. Whenever the automated push is tapped and the app opens as a result, the event would fire, and it would be tracked as an event conversion.