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I am not sure how to answer your question, but let me make the following comments, so you can see where my thinking is going.
The most common usage of density operators in ordinary quantum mechanics is, in the CH perspective, that of assigning probabilities to a quantum sample space at a single time, that is, to a projective decomposition of the identity (PDI). For example, in the formula Tr[rho P], where rho is the density operator and P a projector, a probability is assigned to the property represented by P; more generally, when sum_j P_j = I, one has Pr(P_j) = Tr[rho P_j]. So just as in standard probability theory one has a probability distribution which is be distinguished from an ‘event’ (to which a probability is assigned), I consider rho to be ‘like’ a probability distribution, but because one is often interested in various incompatible PDI’s, and rho can be used to set up a probability distribution on any one of them, I call rho a “pre-probability”.
Consistency conditions refer to a collection of histories, corresponding to some PDI on the history Hilbert space, and the issue is assigning a probability distribution using the generalized Born rule. The consistency conditions limit the sorts of families of histories (of a closed system) to which probabilities can sensibly be applied in this manner. You may be asking whether in place of one (or more) of the properties at a particular time one can use a pre-probability rho, and my immediate response is that I don’t see the physical motivation for it, why it would make sense. The classical analog would be that in some stochastic process in place of a state j at some time, to which the process will assign a probability, you want instead to employ a probability distribution. I cannot say that this is a silly idea, but I would want to know more about what physical process you are trying to model in this way. And the same thing in the quantum case. Is there something to be gained, and if so what?
Some years ago David Mermin published stuff in which he treated a density operator as something like a physical property, and I suspect there are many others who have similar ideas. But I myself have never found this helpful.
I hope the above goes some way towards addressing your question.