*The* open problem in General Relativity? 
Q. Is there a single, clear mathematical question that has emerged as
  the open problem in General Relativity?

I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,

"Die Feldgleichungen der Gravitation," Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847. (Wikisource:The Field Equations of Gravitation.)


         


         

Einstein's notebook, Riemann curvature tensor.


It is difficult to surpass
Willie Wong's thorough 2011 answer
to
the MSE question,
"Open problems in General Relativity."
But perhaps a core mathematical question has risen to the fore since?
Naked singularities? Beyond the Cauchy horizon?
"Electrodynamics of moving bodies?"
Florentin Smarandache's 2013 Unsolved Problems
in Special and General
Relativity (PDF download.)?
Or perhaps Willie's inventory cannot be condensed or sorted further at this point in time?
 A: Perhaps the cosmic censorship conjecture (the absence of singularities outside event horizons) is the most compelling, at least that is what is argued by Klainerman in Cosmic censorship and other great mathematical challenges of general relativity, with reference to Hilbert's requirement that a great problem in mathematics "should be clear and easy to comprehend, difficult yet not completely inaccessible lest it mocks at our efforts. It should provide a landmark on our way through the confusing maze and thus guide us towards hidden truth."

There is no doubt that the cosmic censorship conjecture verifies this
  last criterion, its solution will be a great advance in our
  understanding of general solutions to the Einstein field equations.
  There is also no doubt that it is very difficult. Though young in
  comparison to the other big challenges in mathematics, such as the
  magnificent seven millennium problems, it has resisted so far all our
  efforts and it is obvious to all concerned that a solution is nowhere
  in sight. The conjecture is also clear and easy to comprehend even
  though a completely tight formulation could only be given once a
  solution will be found. It thus only remains to argue, as I will
  attempt here, that the cosmic censorship conjecture is not completely
  inaccessible as it has generated, and will continue to generate, new
  mathematical techniques which allow us to see a glimmer of light at
  the end of the tunnel.

A: Since questions on the interface between GR and QFT are admissible, here's what I consider the open problem in that direction.

Fix a manifold $M$ and consider the set of globally hyperbolic solutions $\mathcal{S}(M)$ of Einstein's equations, possibly with appropriate asymptotic conditions. (a) Give $\mathcal{S}(M)$ the structure or an infinite dimensional manifold (or more general kind of smooth space). (b) Pick a subalgebra $\mathcal{A} \subseteq C^\infty(\mathcal{S}(M))$ that separates points, contains at least the local and multi-local functionals. (c) Show that the choice in (b) can be made such that the diffeomorphism-invariant subalgebra $\mathcal{A}^\mathrm{Diff} \subset \mathcal{A}$ separates the orbits of diffeomorphisms and is closed under Poisson brackets (naturally defined by the variational nature of the equations). (d) Explicitly construct the formal deformation quantization of the Poisson algebra $\mathcal{A}^\mathrm{Diff}$ (say using an infinite dimensional version of Fedosov's method).

The end point of the above construction would constitute a (formal) quantization of GR as a QFT. As stated, there are various choices that can be made in the construction, but if a minimal set of requirements is satisfied, then any such choice is fine. Hence, I consider the above as a "clear mathematical question". In principle, one would also like to replace "formal deformation quantization" in the final step by "strict deformation quantization", but there is not yet a precise consensus on what that means for QFTs, nor is there a constructive method of obtaining strict deformations (unlike formal ones).
