This question was triggered by a comment of Peter Shor's (he is a skeptic, it seems.) I thought that the holographic principle and AdS/CFT dealt with that, and was enough for Hawking to give John Preskill a baseball encyclopedia; but Kip Thorne is holding out, as are others. Why?
It is a matter of opinion, largely dependent on what you mean by settled:
Is information lost or is evolution unitary? There are arguments based on AdS/CFT which make it almost certain that the evolution of a system from pre-collapse matter to after collapse radiation is unitary. This is because there is a dual description - an equivalent description of the system using different variables - in which unitary evolution is automatic. If this is true for black holes in AdS, it is hard to believe it is somehow different for black holes in flat space which are for all intent and purposes as close to their AdS cousins as we wish them to be. Based on that, most people I know are convinced what the right answer is.
How is the information preserved, what is the mechanism for the unitary evolution? In particular, what's wrong with Hawking's original arguments to support information loss? I think it is fair to say that we don't know the answer, at least we don't know how to phrase the answer in the original, gravitational, language. I think there is a lot to learn by making this answer, even if we don't dispute it, as explicit as possible.
As a subset of that, were the arguments given by Hawking, based on Maldacena's superficially similar arguments, convincing? Lots of people are skeptical, including myself. There are some technical and conceptual assumptions that don't seem quite right in that proposed solution.
Susskind complementarity has not solved nor settled the problem. Susskind (first paper considereding firewalls correct) and Harlow (with a loophole) withdrew theirs papers. (later Susskind, posted an amending proposition, but not conclusive)
Polchinsk, et al.
"We argue that the following three statements cannot all be true: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy eective eld theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. Perhaps the most conservative resolution is that the infalling observer burns up at the horizon. Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic distances from the horizon."
...It is widely believed that an external observer sees this information emitted by complicated dynamics at or very near the horizon, while an observer falling through the horizon encounters nothing special there. These three properties | purity of the Hawking radiation, emission of the information from the horizon, and the absence of drama for the infalling observer | have in particular been incorporated into the axioms of black hole complementarity (BHC)...
...There would be an inconsistency if one were to consider a large Hilbert space that describes both observers at once. Such a Hilbert space appears when quantum gravity is treated as an efective field theory, but it cannot be part of the correct theory of quantum gravity if BHC holds...
...that it uses the naturally produced Hawking pairs rather than introducing ad- ditional entangled ingoing bits. This leads us to a rather dierent conclusion, that the thermalization time does not protect us from an inconsistency of BHC....
then throw the second (information is lost, non unitary evolution) or the third proposition (no drama infalling observer), in any case complementarity is not enough.
I would expect the resolution of this dilemma to hold no relation whatsoever to AdS/CFT, being it specific to anti de Sitter spaces, which makes it irrelevant to our universe (unless someone proves an analog version for de Sitter spaces, which seems unlikely)
I think the key to the resolution of this dilemma is the false assumption that quantum evolution is unitary everywhere and for all observers.
To explain why the above assumption is false, take for instance the process of measurement of quantum observables. Regardless of the philosophical doctrine of measurement one subscribes to, the fact is that part of the information of the measured state is lost forever to the party that obtained information from the measurement. Yes, when observers do measurements to quantum systems, the global superposition of the system observer-system is still there "somewhere", evolving under perfect unitary evolution. However, some of the information from the measured system that is accessible to the observer is lost forever, even in principle, because of apparent collapse. So, it should be clear in this case that even if evolution is unitary for outside observers, physical observer eigenstates involved in the measurement will observe violations of unitarity.
so, unitarity is clearly not an absolute property of a system evolution, but depends on the "frame" where it is observed, and how the system couples to the observational frame.
Let's speculate how to apply the above in the scenario of the black hole information paradox: an observer outside the black hole sends information inside it, when the information crosses the event horizon, the information is lost forever for him.
However, observers inside the event horizon still see the whole system behaving under unitary evolution; the information sent inside the event horizon is still reachable to them. In particular, all entanglement correlations between states in and out of the horizon are preserved from their point of view, of course, as long as the measurement devices keep sending data inside the horizon for these observers to measure the correlations.
The trouble arises because some people is naively expecting quantum unitary evolution to be an absolute property that is agreed on by all observers, but quantum measurements are a clear example how this is not true in general.