So far (from points -), it looks like BLD essentially involves the following things (content-wise): (i) the state offering or being in a good position to offer, to each under its power, a justification of its power and (ii) the justification or justifications being such that the interests of each person under the power of the state are adequately addressed (however we are to fill in ‘adequately’).
(4) BW (pp. 4-5): BW considers the case of a coercive state regime in which some, but not all, of those under its power are “radically disadvantaged” (RD) in that they have more to reasonably have much more to fear than others with respect to “coercion, pain, torture, humiliation, suffering, death.” Assuming that the RD are aware of their situation, the state is not in a position to offer them the right sort of justification of its power. Apparently, this is because the state has failed them in the primary task of politics (PT) and they know this.
This case is curious because the state here fails both its primary task and to meet BLD. It would have been better to separate these two conditions. Perhaps BLD would be met, but the condition of solving PT would not, if the RD believed some fairy tale justifying their condition and the state were ready to recite this fairy tale upon arrangements being questioned or objected to? We might also imagine PT being satisfied for this group, but their mistakenly believing that this is not so, in which case the PT but not the BLD necessary condition for legitimacy would be met.
The case is also curious because the comparative property of disadvantage, as far as I can tell, does no work. What matters is that the state has failed PT with respect to RD — and that, because this is known by RD, the state also fails BLD. This fits well with another curious element of the case or RD’s description of it, which is that moral considerations of unfairness in treatment and the like make no appearance. It seems that, for BW, the sort of justification that would work to satisfy BLD need not have any such moral terms or reasons in it. In this, his justify-to condition (for legitimacy) is different from that of the Rawlsian political liberals. On this point, the Rawlsian political liberals strike me as correct.
BLD is also different, at least so far, from the political liberal position on legitimacy in that the focus is on adequate address of private interests, not adequate address across epistemic and evaluative difference — especially difference in moral outlook — in the same society. I think BW will address how differences in moral and other beliefs impact meeting the “justify to” or BLD condition — with respect to other societies and evaluating societies at great historical and evaluative and institutional distance — in his discussion of how relevant sorts of justifications need to “make sense” to those that they are addressed to. But that is to come. This element makes no appearance in this case.
In sum, though the RD case raises some interesting issues that are worth thinking about, it does not say much additional that would help bring further into focus just what sort of justification is required for BLD. Except perhaps that the ‘adequate’ of ‘adequate justification in terms of the agent’s interests’ need not to be filled in with any reasons of fairness (of impact of state coercion on private interests).