If public reason liberalism is simply a negotiator and mediator, then--as Brian Kogelmann points out on a Facebook discussion--it shouldn't always end up with a liberal position or outcome. Rather, the compromise position it works out should depend upon the particular values of the people doing the compromising, including what they are willing to concede and on what terms.
It is instead more like a very opinionated therapist. Imagine two spouses aren't getting along because one wants to raise the kids Randian and the other wants to raise them Marxist. The therapist instead tries to convince them that the reasonable, respectable position is...the therapist's own favored way of parenting. And then, no matter which parents the therapist deals with, the therapist recommends basically the same methods every time. Here, it seems like the therapist simply has a substantive view of parenting. They say, "We should compromise" but mean "You should agree with me."