In short, they mostly seem far-fetched to me, probably due to a superficial reading of the paper (as Mitchell_Porter admits). For example:

I also noticed that the authors were talking about "Fisher information". This was unsurprising, there are other people who want to "derive physics from Fisher information"

The Fisher information in this paper arises automatically at some point and is only noted in passing. There is no more derivation from Fisher information as there is from the wavefunction.

they describe something vaguely like an EPR experiment ... a similarly abstracted description of a Stern-Gerlach experiment

The vagueness and abstraction are required to (1) precisely define the terms (2) under the most general conditions possible, i.e., the minimum information sufficient to define the problem. This is completely in line with Jaynes' logic that the prior should include all the information that we have and no other information (the maximum entropy principle). If you have some more concrete information about the specific instance of Stern-Gerlach experiment you are running then by all means you should include it in your probability assignment.

They make many appeals to symmetry, e.g. ... that the experiment will behave the same regardless of orientation. Or ... translational invariance.

Again, a reader who is familiar with Jaynes will immediately recognize here the principle of transformation groups (extension of principle of indifference). If nothing about the problem changes upon translation/rotation then this fact must be reflected in the probability distribution.

hope that some coalition of Less Wrong readers, knowing about both probability and physics, will have the time and the will to look more closely, and identify specific leaps of logic, and just what is actually going on in the paper

• in fact this is what I was trying to do here.