I've been commenting on the odd sociology of the public justification theorists. The striking feature of their work is that they almost never engage in public justification. Now, I recognize that their work on the epistemology of public reason might not need to be accessible or publicly justified--that depends on whether the theorist in question believes that the principle of public justification applies to itself or not. But oddly, you don't see public reason theorists doing what they take themselves to be doing--they basically never try to show that the liberal order is accessible and defensible in light of the reasons slightly idealized people have, and they don't talk much to the public. Instead, they try to convince each other than their theory justifies liberalism.
How might public reason liberalism go if the public reason liberals (PRLs) stuck to the motivations behind their project? Something like this, I think:
First, you'd see the PRLs survey, say, 20,000 or so randomly selected people from all one the world to determine their values. These carefully crafted surveys would need to include questions which demonstrated the relative weight people assign to various principles, including what trade-offs they are willing to make.
You'd also want to do robustness checks. A problem with such surveys is that people don't like to say they don't know, and many of their political positions are just parroting what their neighbors happen to say at the time. So you'd want to administer the same surveys (with questions randomized in order) to the same people over and over to ensure they actually believe what they say. If we get the results that Converse got in his studies (see Converse 1964), where there was basically zero correlation among subject's responses between surveys, we'd suspect as Converse did that they were innocent of real opinion.
Further, we'd probably want to operationalize our conception of "reasonable". We might do various psychological exams to see if the people can follow basic chains of evidence, or are subject to severe bias, and so on. We could even presume that "reasonable" = "agreeing with John Rawls" and test to see who accepts his ideas and who doesn't.
We'd probably want to collect demographics, too, to see how that affects opinion.
With all this in place, we could then determine whether, for instance, reasonableness is correlated or anti-correlated with various views, all while controlling for confounds. We could statistically determine, using enlightened preference methods, whether reasonableness as an operational variable leads to various kinds of convergence or not. Most importantly, we could then see what range of values reasonable people have and actually determine what their values converge on, what their values reject, and so on. Because our surveys give weights to their values and their trade-offs, we could then determine which sorts of compromises and the like they would in fact make.
You might then expect some "experimental philosopher" PRLs to try to convene various kinds of deliberative fora in which real citizens (perhaps those who qualify as reasonable by our operationalized definition) are brought in, and the theorists try to convince them of the conclusions which we revealed to be justified by public reason.
What we probably wouldn't want to do, if we cared about public justification, is create a largely insular academic community where we make relatively standard liberal philosophical arguments and standard social scientific arguments for this and that, but then specify that we regard our project as distinctly different from what other liberals and other social scientists are doing.
UPDATE: Maybe this is what Klosko was up to here: https://www.jstor.org/stable/pdf/2939045.pdf?fbclid=IwAR267eIhOwyDlBA16nZBao7NPYVtrHjqYUllYtleDzaHngo2L8397Z7PE9I&seq=1