It's surprisingly difficult to find a definitive answer to this; couldn't find it mentioned in the Apple documentation and couldn't find a definite yes/no after searching past questions.
The question is simple - if the app requests a background fetch to be performed after N time, then the user terminates the app. Will the OS still launch the app into the background to perform the background fetch?
Okay, once again background modes cause confusion. No offense to the other people trying to help, but this is more complicated than it seems.
First of all: This is out of date, as Sausage guessed in the comments. I know this for a fact, because the section about VoIP apps is still explaining the "old way" to do this, with a handler that gets called periodically. I investigated this a bit for this answer, so I suggest you go and read that. The important lesson for this case here is that iOS makes a distinction between an app being terminated by the user or by the system, plus it also plays a role whether the phone was rebooted or not.
So to sum this (and your question) up you basically want to know whether this part of the above, outdated documentation is still correct word for word:
I thoroughly investigated the rest of the docs, but did not find any definite answer, so it unfortunately boils down to what dan already suggested: Test it. My gut feeling is that the documentation is still correct in that regard, though (as said what's not is the VoIP stuff). I say that because the UI in the Settings app calls the feature "Background App Refresh", so users are probably supposed to understand that an app having this permission won't refresh when they "push" them out of background (i.e. home button -> swipe it out). For regular users, apps are either quit (not in the task manager at all), in the foreground (using them) or in background (they're in the task manager and another app is in foreground and/or the phone is locked).
To really test this you'd have to write an app and actually carry it around a bit (I assume at least two days) in each condition. First while it is in background (the OS should periodically let it fetch, as you probably know this can also be triggered in Xcode) and then while it is force-quit. The problem is to verify that it fetched stuff. I'd go with a logfile that can be shared via iTunes. I have typed up some code for this:
This would go into the app delegate, and don't forget to add the
Application supports iTunes file sharing
boolean setting in your app's plist. I will leave this running on my development device for a bit and check the logfile, eventually reporting back here. Feel free to test it yourself, too.Reading the Apple documentation here I found this text snippet which should explain your question:
The second option is exactly about downloading the data, which can be delegated to the system even if the can be terminated.
EDIT:
https://devforums.apple.com/message/873265#873265 (login required)
That post was by an Apple employee so I think i can trust that this information is correct.
OLD answer:
According to this answer wrote by a top user: iOS background fetch: your app won't be woken up again.
It really doesn't make sense for it to be woken up...it kinda invalidates the user killing the app.
Having that said there are different ways a terminated/force quit app can be launched again:
Location updates either through use of region monitoring or the significant-change location service. See this answer and make sure to read this entire page from Apple docs.
Rebooting the device would also undo anything blocked through force-quit