> The Dirac equation and Quantum Electro Dynamics (QED) unify quantum mechanics and special relativity

And more generally the Standard Model, which includes the weak and strong interactions. The SM is a quantum field theory, which, as you say, unifies QM and SR.

> (non-accelerating frames of reference).

No, SR and QFT are not limited to non-accelerating frames. They are limited to small enough regions of spacetime that spacetime curvature is negligible. This experiment is an illustration of that: it compares an accelerated atom with a free-falling atom to show the phase shift between them, and the lab frame in which it is done is accelerated--but the SM and SR work just fine. But the experiment does not show any effects of spacetime curvature.

> the remaining piece is either to extend QED/QCD to accelerating frames of reference

No, that's already done. See above.

> or to quantize general relativity.

That's the big missing piece, yes. We know how to write the QFT of a massless spin-2 field (which is our naive expectation of what a QFT for gravity would look like), and we know that the classical limit of that QFT is the classical GR we have now. But we know that QFT has to be just an effective theory, just like the Standard Model; it can't be the final answer.

> That would likely predict the phase shift observed in this experiment.

The theories we already have (Standard Model + the equivalence principle are all we actually need) are sufficient to predict that. Of course any more comprehensive theory will have to reproduce that prediction, yes.