By default iPhone will disconnect idle application from server to save battery. But we will have push notifications in such period. You will be informed when you receive new message.
But also right now in output messenger, we are doing that only when you are offline in other devices too. Suppose if you have left login in Output Messenger Desktop client, the message will be delivered only to desktop & no push notification will be sent to your mobile. If all other clients are disconnected, then Output Messenger server will send push notification to your mobile device.
We are working right now to send mobile notifications for idle, Stepped Out & lunch status too.Hope in 2 weeks time, we will release the updated version.
For now, you can make sure to exit the desktop client while you are moving out of the office.