When we measure an observable A of a quantum system, we get an eigenvalue of A. Without worrying about connotations of Copenhagen vs. MWI, etc., let's just call this "collapse."
Question: Among people who are not convinced that decoherence gives a complete answer to this problem, what are the strongest objections that have been raised against its acceptance?
A somewhat cryptic take on this seems to be given in this 2013 paper by Bubb, who says on p. 20,
The argument here is not that decoherence provides a dynamical explanation of how an indefinite quantity becomes definite in a measurement process—Bell  has aptly criticized this argument as a ‘for all practical purposes’ (FAPP) solution to the measurement problem. Rather, the claim is that we can take the decoherence pointer as definite by stipulation , and that decoherence then guarantees the objectivity of the macroworld, which resolves the measurement problem without resorting to Copenhagen or neo-Copenhagen instrumentalism.
The reference is to a 1974 essay by Bell, "On wave-packet reduction in the Coleman-Hepp model," which I guess dates to long before people started talking about decoherence. The Bell paper can be found online (presumably illegally). It talks about a certain toy model, not about decoherence. The point he seems to be making is that it matters that the "collapse" described by this toy model only happens in the limit $t\rightarrow\infty$. If I'm understanding correctly what Bubb is saying, then I guess the idea would be that in decoherence, the off-diagonal elements of the density matrix fall off exponentially, but they never actually hit zero. If Bubb is saying that he considers this to be the strongest objection remaining to decoherence as an explanation of "collapse," then it would seem to me to be an extremely weak objection.
Please limit answers to the specific question asked above. I don't want to open up a can of worms about other questions such as whether some particular interpretation of quantum mechanics can/should/does/doesn't "explain" the Born rule. I'm also not interested in having decoherence explained to me. I think I already understand it at the level of Joos and Zeh, The emergence of classical properties through interaction with the environment, Z Phys B 59 (1985) 223, which should be sufficient for the present discussion. I'm also not interested in hearing people give yes/no answers as to whether decoherence does explain collapse, because obviously that will never lead to a conclusion that everyone accepts. I'm just asking for an outline of what reasons people do offer for objecting to this as a proposed mode of explanation. Also, I'm interested in whether my interpretation of Bubb's remark is correct.