Background
Decoherence solves some aspects of the Measurement Problem by ensuring that at the end of a measurement process off diagonal terms in the density matrix are supressed. However decoherence does not solve the Problem of Outcomes: how to go from a diagonal density matrix to a single definite outcome for an individual measurement run.
The solution in the ABN Paper
The solution to the Problem of Outcomes in the 2013 paper by Nieuwenhuizen et al ( Scientist propose solution to quantum measurement problem, explain further) adopts a Statistical/Ensemble interpretation of QM.
Consider a system S (with eigenvalues $s_i$ of some observable) interacting with a measuring aparatus A (with pointer states $A_i$ which can correlate to the $s_i$).
We start with a density matrix before measurement of
$$\hat{D}(t_0) = \hat{r} (0) \otimes \hat{R}(0)$$
After the physical measurement process (including decoherence to remove off-diagonal terms) we end up with
$$\hat{D}(t_f) = \sum_i p_i \lvert s_i \rangle \langle s_i \rvert \otimes \hat{R}_i$$
They call this truncation and registration.
The density matrix above applies to a complete ensemble $ \mathcal{E} $ of the S+A system (representing multiple runs of the same experiment with a distribution of outcomes given by the $p_i$).
They then argue that after the measurement (where we now have an objective outcome $A_i$ say shown on the measuring device) we can now retrospectively restrict to a subensemble $ \mathcal{E}_i $ : for this restriction the density matrix reduces to just
$$\hat{D}_i(t_f) = \lvert s_i \rangle \langle s_i \rvert \otimes \hat{R}_i$$
Hence each experiment has a unique (albeit unpredictable) outcome: Measurement Problem solved!
My question
To justify the final reduction step they claim they need to first demonstrate something they call hierarchic structure (page 15):
We will rely on a property of arbitrary subsets of runs of the measurement, their hierarchic structure. Namely any subset must be described at the final time by a density operator of the form $\sum_i q_i \hat{D}_i $ with arbitrary weights $q_i$.
It seems from this that the full solution to the Measurement Problem hangs on a rather subtle technicality: can anyone either expand on the relevance of this hierarchic structure property or refer to other papers where this property has been used. I am trying to find out if this approach has been generally accepted or if it has been refuted before studying the paper in more detail.
