Sorry, I was still talking ("this problem") about the example I introduced.

If the method works on other problems, that seems like good evidence it works on your specific conference paper problem, no?

Which recommends sharing "average estimates plus justifications" and "provide the mean or median estimate of the panel plus the rationales from all panellists".

Indeed, it does - but it says that it works better than purely statistical feedback. More information is often better. But why is that relevant? You are moving the goalposts; earlier you asked:

It is not clear to me how any decision method that does not allow for sharing of evidence can yield the right answer for this example.

I brought up prediction markets and Delphi pools because they are mechanisms which function very similarly to Aumann agreement in sharing summaries rather than evidence, and yet they work. Whether they work is not the same question as whether there is anything which could work faster, and you are replying to the former question, which is indisputably true despite your skepticism, as if it were the latter. (It's obvious that simply swapping summaries may be slower than regular Aumannian agreement: you could imagine that instead of taking a bunch of rounds to converge, one sends all its data to the other, the other recomputes, and sends the new result back and convergence is achieved.)