There are irreconcilable differences between general relativity and quantum mechanics. A lot of thought has gone into the problem, with little success. But even if someone did come up with something concrete, what then? Try to come up with an interesting experiment that combines gravity and quantum mechanics.

I know of only one, and whether it tests anything at all depends on which interpretation of QM you have. Based on a Geiger counter, either place, or don't place a heavy weight. Try to measure its gravitational pull regardless of what you do. It only measures a pull if you placed the object. If you believe in the Everett interpretation, this says that gravity, at least to a first order approximation, splits with the universe. We do not have sensitive enough instruments to measure non-linear differences from GR.

History tells us that theoretical science done in the absence of experiment is unlikely to lead to useful knowledge.

What is this experiment called? Any links to read up on?

perhaps reading this http://www.hedweb.com/everett/everett.htm

(especially question 7) will help, even though it does not give the name for such an experiment.

I once saw a lecture by Freeman Dyson where he said that if one were to build a graviton detector with the cross-sectional area of the Earth and point it at the sun for the age of the Earth, the number of gravitons one would expect to detect:

Four.

So, whether there's science still to be done and whether we'll ever be capable of building apparatus that can actually test it are two different questions. For instance, what if the next interesting thing post-Higgs Boson happens at energies 1000 times bigger? There's a good chance we'll never build an accelerator that powerful.

We may not need a more powerful accelerator. Remember: The cosmic rays contain much more energetic particles than that can be produced in accelerators so far.

It is possible that we will discover new phenomena and new ways to test gravitational theories once we can observe gravitational waves. I expect we will detect gravitational waves in ~10 years and identify a specific source in ~20 years; sooner, if there are some powerful sources that we did not think of yet.

As someone who does astrophysics right now, trust me, we want to wring everything we can out of accelerators before we start trying to use cosmic objects to probe the laws of physics.