What are the weaknesses, if any, of the relational interpretation of quantum mechanics? Carlo Rovelli's 1996 relational interpretation of quantum mechanics (RQM) seems to solve many of the quandaries of traditional theories, including the Copenhagen interpretation (what privileges the observer? Why does he/she instigate wavefunction collapse); the many-worlds interpretation (an infinity of unobservable universes); and spontaneous collapse theories such as that of GRW.
However, there seems to have been limited academic discussion of the theory. Is this due to a shortcoming of the theory, or has there simply not been enough time for it to disseminate?
 A: I personally consider RQM to be an attempt at the "best of both worlds" between Copenhagen and MWI. By making measurements observer-dependent, it essentially accomplishes the same thing as entanglement of the system and observer in MWI. By refusing to speculate on other worlds, it has the veneer of realism of Copenhagen. (Full disclosure: I am an MWI supporter but am open to all ideas.)
The problem with RQM as far as I can tell is the "sparse ontology" or "flash ontology" problem. It is discussed briefly here. All measurable quantities in RQM are only defined in the exact instant that they are measured; for most instants in time, the state of the system is essentially undefined. The fundamental reason for this is that RQM postulates that the only thing you can talk about in physics is how systems interact rather than how they are.
According to your taste, this may be a positive or negative aspect of the theory.
For me it definitely seems unphysical at best and like borderline solipsism at worst.  But I can see obvious counterarguments:

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*All interpretations of quantum mechanics have something unphysical about them.

*Pretty much all advances in physics since Copernicus have required us to get used to a new counterintuitive idea about reality, and this is probably no greater or worse than the previous ones.

A: It's fair to say that the discussion about RQM has begun to gain momentum in recent years. I have the impression that it has increased in popularity among physicists and philosophers of physics.
But to make contact with the title of your question, there certainly are substantial open problems. Inter alia, some critics say that it leads to solipsism, and that it suffers from a sort of preferred-basis problem. It's also slightly unclear how the story extends to more general interaction settings (say, continous interactions). If you're interested in diving a littler deeper into some of these issues: here and here are some recent critiques, which Rovelli has replied to here and here, respectively.
