Semi-classical gravity is not considered fundamental, yet it escaped to experimental falsification. Of course, maybe we don't have yet the technology, or at least ideas of experiments we can do, to falsify it. It would be nice to be able to differentiate it experimentally from various quantum gravity approaches. But theoretically, it stands pretty well: being the most straightforward union between general relativity and quantum mechanics, it inherits their successes.

Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?

One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).

Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).

About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here, here, and at arxiv:1309.2309.

I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.

Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?

One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).

Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).

About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here, here, and at arxiv:1309.2309.

I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.