I just did same test on my Nexus 5x (6.0.1 stock), with your settings (scheduled sync off, push on).
2016.03.13 13:48:14.011 -0400 PushConnectivityReceiver [][][][][] onReceive ConnectivityManager.CONNECTIVITY_ACTION: ni = [type: WIFI[], state: CONNECTED/CONNECTED, reason: (unspecified), extra: "Despotica", roaming: false, failover: false, isAvailable: true]
2016.03.13 13:48:14.012 -0400 PushConnectivityReceiver [][][][][] Currently active network info: [type: WIFI[], state: CONNECTED/CONNECTED, reason: (unspecified), extra: "Despotica", roaming: false, failover: false, isAvailable: true]
2016.03.13 13:48:14.013 -0400 PushConnectivityReceiver [][][][][] Scheduling the delayed alarm
2016.03.13 13:48:14.014 -0400 AlarmCompat_api19 setWindow: 0, 1457891299013 (2016-03-13 13:48:19) 5000, PendingIntent{4ea362b: android.os.BinderProxy@bab7e7a}
2016.03.13 13:48:19.045 -0400 PushConnectivityReceiver onReceive: intent = Intent { act=org.kman.AquaMail.ACTION_CONNECTIVITY flg=0x10000014 cmp=org.kman.AquaMail/.core.PushConnectivityReceiver (has extras) }
2016.03.13 13:48:19.046 -0400 PushConnectivityReceiver [][][][][] Starting the service to start watchers
This is the "delayed alarm" above.
In your log, the line where the app sets this alarm is present:
2016.03.13 13:48:14.013 -0400 PushConnectivityReceiver [][][][][] Scheduling the delayed alarm
2016.03.13 13:48:14.014 -0400 AlarmCompat_api19 setWindow: 0, 1457891299013 (2016-03-13 13:48:19) 5000, PendingIntent{4ea362b: android.os.BinderProxy@bab7e7a}
but the alarm does not go off (does not trigger):
2016.03.13 13:48:19.045 -0400 PushConnectivityReceiver onReceive: intent = Intent { act=org.kman.AquaMail.ACTION_CONNECTIVITY flg=0x10000014 cmp=org.kman.AquaMail/.core.PushConnectivityReceiver (has extras) }