what does differential privacy (in machine learning) promise or guarantee?
Model inversion attacks are, by definition, impossible if the model generation process is $\epsilon$-differentially private for a sufficiently small $\epsilon$: differential privacy guarantees that if your original data was completely different, you would not be able to tell the difference just by looking at the model. Your data is effectively indiscernible from noise.

That said, the strength of the guarantee depends on the value of $\epsilon$, you can read a short explanation of that on this other question. If $\epsilon$ is very large, model inversion attacks might be successful in retrieving noisy or probabilistic information. I'm not aware of any attempt to experimentally verify for which parameters of $\epsilon$ it starts becoming a problem.