Hi Rob,
Yes it should also start your app when the app is not running or suspended, but it depends on heuristics of how much the user uses your app also something to keep in mind is that:
If the user rebooted his device and never launched the app since the reboot, the app will never wake up remotely and if the user killed the app manually from the app switcher, the app will also never wake up remotely.
I created a small app similar to your use-case where I do a http request when the content-available flag is set:
onNotification: function(event) {
if (event['content-available'] === 1) {
ajax({
url: "http://192.168.0.30:8888/index.php",
dataType: "text"
})
.then( function( result ) {
console.log(result);
push.setContentAvailable(push.FetchResult.NewData);
});
}