Existence of a Trapped Surface to the Existence of a Black Hole Does the existence of a trapped surface in a region of space (not necessarily either of the vacuum or symmetric spacetime) indicates (theoretically) the existence of a "black hole" there? And if yes, is it sufficient to describe the existence of a "black hole" (at least theoretically) by the existence of a trapped surface there?
Note: Penrose, in his famous paper in 1965 ["Gravitational collapse and space-time singularities". Phys. Rev. Lett. 14 (3): 57–59] claimed that [Source: Wikipedia],

A trapped surface is one where light is not moving away from the black hole.

But it is also well known that singularity can be avoided in the quantum treatment of such problem. Now the notion of singularity is eternally related to the concept of  a trapped surface!
Along-with the previously cited references, I also had a look into the following references, but unfortunately my doubts remained uncleared (may be I've missed something there, which can clearly point out and answer my doubts):

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*"Lecture Notes on General Relativity" by Matthias Blau, Albert Einstein Center for Fundamental Physics, Institut f¨ur Theoretische Physik, Universit¨at Bern. (Chapter 29, Section E; updated version of October 8, 2022)


*Senovilla, Jose M. M. (September 15, 2011). "Trapped Surfaces". International Journal of Modern Physics D. 20 (11): 2139–2168. arXiv:1107.1344 [gr-qc].


*Hawking, Stephen & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. (Preface, page $7-8$ and Chapter 8, page $266-267$; reprinted edition of 1994)


*Bengtsson, Ingemar (December 22, 2011). "Some Examples of Trapped Surfaces". arXiv:1112.5318 [gr-qc].


*Kanai, Yuki. Siino, Masaru. Hosoya, Akio (August 3, 2010). "Gravitational collapse in Painlevé-Gullstrand coordinates". arXiv:1008.0470v1 [gr-qc].
So what's the way out? How the existence of a trapped surface and the existence of a black hole is (theoretically) connected to each other?
 A: 
Does the existence of a trapped surface in a region of space (not necessarily either of the vacuum or symmetric spacetime) indicates (theoretically) the existence of a "black hole" there?

Not necessarily, without further assumptions on the spacetime geometry.
If we restrict ourselves to solutions of EFEs with matter satisfying weak or strong energy condition, then trapped surface ($T$) must be inside the black hole region of spacetime, $T\subset B$ (see Wald, proposition 12.2.2).
However if we drop constraints of EFEs then it is possible to have a trapped surface without a black hole (or alternatively a black hole without a trapped surface).
For example, consider the following depiction of a  “bouncing” spacetime (image taken from this paper):

This figure represents the lightcone structure (in $t-r$ plane of some reasonable coordinate system) of a spherically symmetric object (its outer surface is the thick line) that undergoes collapse and subsequent bounce. When the radius of the object decreases below its Schwarzschild radius, trapped surfaces appear (trapped surface would be a point in this figure). But nevertheless, there is no causally disconnected region in this spacetime, no event horizon and so no (true) black hole.
Of course, such a spacetime would not satisfy Einstein equations with a “normal” type of matter. But spacetimes with such features are considered as models of various quantum gravitational effects (such as black hole to white hole transitions). Often, the term “black hole” is still used for such spacetimes even though they no longer satisfy the classical definition.
