No. Basically even though you have an open-segment between EZE-AKL, you do have one year to fly that segment. Which is your right.

However as QF is cancelling the service, and if you choose a date which you want to fly on, well, what are you going to do?

QF should reroute you and NOT charge you as it is not your doing. If say that EZE-AKL still exists and you're rerouting, then it's a different story.

I'd try to speak to a sales rep about this ...