What are the (philosophical) implications (if any) of this experiment in relation to the Bohr-Einstein debates and hidden variable theories? What are the (philosophical) implications  (if any ) of  this   experimental result, in relation to the Bohr - Einstein debates and hidden variable theories of quantum mechanics? A related question can be found here
 A: There seems to be a confusion in this experiment, mixing frameworks and models.
In the publication   in the beginning of the description of the experiment they state:

"First, we developed a superconducting artificial atom with the necessary V-shape level structure " .

Super conductivity is a meta level , emergent from the underlying quantum mechanics, and modeled with quantum mechanics. That it can be modeled quantum mechanically does not subtract from the fact that it  is emergent from zillions of particles. My opinion is supported by the statement in this tutorial on quantum trajectories , that everything is within the scope of standard quantum mechanics and its postulates.

"In these systems, the stochastic equations arise as effective evolution equations, and are in no sense anything other than standard quantum mechanics(except, perhaps, in the trivial sense of approaching the limit of continuous measurement)

One cannot argue for the basic postulates of quantum mechanics on emergent data from underlying quantum mechanics, imo. In a sense it is like taking the mechanics in a video game to deduce the mechanics in real three dimensional space.
Let me give an example from classical physics:
Thermodynamics emerges from classical statistical mechanics, can one deduce from thermodynamic states the underlying particle interactions and assert their continuity and change classical mechanics postulates?
It so happens that the emergent behavior of electron pairs in metals can be modeled quantum mechanically ,  that means that the theory fits the data with postulates appropriate to the data. If for this superconducting construct, "atom", there is a discrepancy with the basic quantum mechanical postulates, too bad, it means that the model's postulates  have to change for superconductors, with whatever this implies. 
In my opinion to really question the underlying quantum mechanical framework of elementary particles one needs experiments with elementary particles, not with collective phenomena emergent from zillions of such particles. Hidden variable theories for the basic quantum mechanics framework  cannot be based on emergent states.
