Is the black hole interior a part of objective reality? Lately I have begun thinking about black holes more than back when I first learned (admittedly, in a hurry) about general relativity. Of course, I already made peace with observer dependence at the stage of special relativity. But that does not necessarily force one to give up objective reality, despite the well-known apparent paradoxes, which all have kind of reasonable explanations.
On the other hand, the more I think about black holes, the more I feel that they are an imposition on any belief in objective reality much in the same way as quantum field theory is.
To be more specific: information about matter falling into a black hole gets lost for the outside observer (Edit: I actually wanted to focus more on the impossibility of getting information about what happens inside the hole, which is a little different from historic information, at least if one allows for some indeterminism), and this information loss happens in an infinite duration. On the other hand, the co-moving observer of the free-falling matter reaches the event horizon in finite time, and after that other things keep happening. It is not just one ageing faster than the other, like in the twin paradox, which can be considered compatible with objective reality. Instead it is completely another reality happening after the "reference reality" has ended, and no careful observer can ever falsify the things believed to happen after the end of his/her universe.
This reminds me faintly of the double slit experiment, where we can not tell where the electron or photon is, unless we impose an interaction that localizes it and hence destroys the prior wave function. It makes no sense that we speculate about the position of the particle before this event, because we cannot observe it without changing the configuration, so we do not consider the particle position of the uncollapsed wave function a part of objective reality.
So is it scientifically justifiable to consider the story happening inside the event horizon of a black hole as a part of objective reality of us outsiders? Are we allowed to claim that although we cannot see what happens inside the black-hole, we can solve Einstein's equations based on what we know about the matter before it fell into the black hole, and conclude from it what happens there? Or does the transition to the interior of a black hole represent such an extreme discrepancy between actual information and emotional information that it is, scientifically, on the same level as fairies and elves?
Note that I am not educated enough in philosophy for more sophisticated arguments about reality. I just would like interpret it in the most naive sense of Ockham's razor/best scientific practices.
Edit (2021-02-03): I think the confusion between me and my cherished responders has been caused by my misunderstanding of the canonical interpretation of general relativity (most probably due to some popular depiction of black hole physics in TV documentaries).
I had the impression that the infinitely red-shifted vision of the matter falling into the black hole on the one hand and the actual matter falling with a co-moving observer on the other hand, were two different versions of reality. Furthermore, because one version takes infinitely long to reach the horizon, and the other one continues after passing the horizon, I have thought that the "reality of what happens inside the hole" was somehow situated in time after the vision from the outside has completed (which means: after "the end of the distant observer's universe").
But I think I understand now, what "really" happens according to the theory. What the co-moving observer experiences is actually happening in parallel to what the distant observer sees. Just because light is progressively red-shifted near the horizon, the distant observer is not able to see what really happens to the co-moving observer after say 5 minutes after passing the horizon, but sees a distorted mirage of the process up to the horizon.
So in analogy to ships cruising the ocean, my misconception was that I concluded if fellow ships vanish behind the earth's horizon, they would cease to exist, while it's only impossible for their light to travel to the observer. Put that way it sounds funny, but it gets easy to fall into that trap because of the involvement of time in the curvature of general relativity.
Does that sound correct?
 A: Unfortunately, your desire to avoid philosophical considerations is on a losing wicket. Specifically, the ontology or nature of the underlying reality is a philosophical issue. The baseline view of science, known in quantum physics as the Copenhagen interpretation, is that you simply ignore such issues as scientifically meaningless and concentrate on the ability of your maths to predict experimental results.
A particular feature of Relativity is that the overall causal structure of objects and events is immutable and therefore, as far as the theory is concerned, is objectively real. What is subjective for each reference frame are details like exact masses, momenta, time periods, distances and directions.
But your ideas are not quite right. Your suggestion that "information about matter falling into a black hole gets lost for the outside observer, and this information loss happens in an infinite duration" is wrong. Hawking has shown that, given sufficient time, the black hole will evaporate and the information will be transformed and regurgitated. By transformed we mean that it will no longer be information about the old infalling matter, but about something more directly related to the current activity of the black hole. Information is never wholly lost.
You also suggest the idea of "a part of objective reality of us outsiders". This is a contradiction in terms. Objective reality is by definition true for all observers, a reality confined to a given class observer is by definition subjective. This issue is of fundamental significance to Relativity theory, no more so anywhere than inside a black hole.
Having said that, one can continue to explore the maths of what we might expect to go on inside a black hole's event horizon. For example Penrose demonstrated the existence of trapped surfaces, which are basically smaller event horizons inside the outermost one. An astronaut could fall through the event horizon of a giant black hole and hardly be aware they had done so, but everything they could see would be falling in from above and there would only be further trapped surfaces below them. Since this is all in principle inaccessible to science, positivists such as Hawking see it as meaningless. However more flexibly-minded physicists such as Penrose see it as a necessary part of the immutable overall reality (whatever those equations might represent), with its inaccessibility being merely subjective. Your choice will not be based on your physics but on your philosophy.
A: 
Is the black hole interior a part of objective reality?

You haven't defined "a part of objective reality," so you haven't asked a well-defined question.
Note that black hole horizons aren't the only horizons we can have in GR. There are also cosmological horizons, and even the horizon of an accelerated observer in flat spacetime. If you have in mind some alteration of reinterpretation of GR to get rid of the philosophical blight of horizons, you need to deal with these other horizons as well.
