Closing a flightplan just means sending an ARR (arrived) message to all its original recipients. We can do that, or somebody else can do it. If someone else does it, we won't usually get the message so we won't know it's been closed, but that's fine. It's only the pilot who needs reassurance that it's been closed.
In the example above the French unit, if they'd looked, could probably have seen that an arrival message had already been sent.