Reading the paper today in order to avoid doing actual work, it seems to me that really all Hobson is doing is to attempt to handwave away the distinction between proper and improper mixtures, and then attaching an ignorance interpretation to the latter. Basically, he starts with an entangled state like
|\Psi_{MO}\rangle=c_1|M_1\rangle|O_1\rangle + c_2|M_2\rangle|O_2\rangle,
which can be interpreted as the post-measurement state of a measurement system M and an object system O that started out in a superposition of its two accessible states |O_1\rangle and |O_2\rangle, with the index of M indicating which state was detected.
This can be described equivalently by the density matrix
\rho_{MO}=|\Psi\rangle \langle\Psi| = |c_1|^2|M_1O_1\rangle\langle O_1 M_1| + |c_2|^2|M_2 O_2\rangle\langle O_2 M_2|+ c_1c_2^\star|M_1 O_1\rangle\langle O_2 M_2| + c_2c_1^\star|M_2 O_2\rangle\langle O_1 M_1|
In order to now get a mathematical object that describes either of the systems on its own, one uses the procedure known as 'tracing out' the other subsystem, that is, one performs the partial trace over, say, the object system to get the description of the measurement system:
\rho_M=\mathrm{tr}_O\rho_{MO}=|c_1|^2|M_1\rangle\langle M_1| + |c_2|^2|M_2\rangle\langle M_2|
Mathematically, this is the same object that one would use to describe a system that is prepared either in the state |M_1\rangle or |M_2\rangle with a respective probability of |c_1|^2 or |c_2|^2. However---and this is where the argument goes wrong, I believe---in case this object is arrived at by tracing out the degrees of freedom of another subsystem, one can't interpret it in the way that the system is in fact in either of the states |M_1\rangle or |M_2\rangle, and we just don't know which. This is the basis of the distinction between 'improper' (arrived at by tracing) and 'proper' (arrived at by epistemic uncertainty) mixtures, due originally to Bernard d'Espagnat, I believe.
Now, Hobson is well aware of the distinction, but---as best I can gather---he believes it doesn't matter, as locally, we can't tell a difference between the two. This much is true. But nevertheless, there is an important and simple distinction (that has direct experimental consequences). For consider now that the state |\Psi\rangle to describe an EPR pair, say, electrons entangled regarding their spin properties, and in the possession of Martha and Oliver (sorry for introducing nonstandard nomenclature here).
Then, both parties would quite reasonably consider the part of the state in their possession to be described by a mixture such as \rho_M above. But, interpreting their state as being actually in a proper mixture of the form---that is, being actually in a definite state they just don't happen to know---has important consequences, not for the outcomes obtained in their local experiements, but for the correlations between these outcomes. For they would judge that the total state must be just the tensor product of their local states, which looks like this:
\rho_{MO}^\prime=\rh<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f635.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt="o_O" title="Er... what? o_O" data-smilie="12"data-shortname="o_O" />\otimes\rho_M=(|c_1|^2|M_1\rangle\langle M_1| + |c_2|^2|M_2\rangle\langle M_2|)\otimes(|c_2|^2|O_1\rangle\langle O_1| + |c_2|^2|O_2\rangle\langle O_2|),
which is a state corresponding to an epistemic mixture of the possibilities |M_1 O_1\rangle, |M_2 O_2\rangle, |M_1 O_2\rangle and |M_2 O_1\rangle---four possibilities, while there were only two in the original state \rho_{MO}, corresponding to the fact that entanglement means that whenever Martha detects the state |M_1\rangle, Olliver detects |O_1\rangle, and whenever she detects |M_2\rangle, Olliver detects |O_2\rangle. In assumind that their local states are epistemic mixtures of two distinct possibilities, that is, that they really are in one of two possible states---the analogy of obtaining a definite measurement outcome---, Martha and Olliver can no longer account for the correlations between their measurements.
Now, in response to this, Hobson appears to put up some handwaving about how it's the correlations that are in a superposition, not the states, but it's unclear to me what that's supposed to mean. And regardless, the problem lies in his assertion that in an entangled state such as |\Psi_{MO}\rangle, one can always say that (in his example) 'one photon measures the other', leading thus to definite states; but for any pair of entangled photons, I can do either of two things: I can measure the photons individually, obtaining definite outcomes; or, I can measure the entanglement of the state, which is incompatible with the assertion that the photons are in definite states (at least in standard QM). The fact that I can make that choice and must use different prescriptions to account for my observations is exactly what the measurement problem is all about; in Hobson's proposal, once two photons are entangled, and one has 'measured' the other, I would never observe any of the phenomena that make quantum mechanics so interesting---superposition, entanglement, interference, etc.
