"Mutual recognition" suggests to me that Locke was on the right track. Acting arbitrarily regarding other people is not taking their persons and interests into account. Recognizing the existence of other people and their interests is the first step towards not acting arbitrarily.

The problem is that to imbue "recognizing" with any further regulatory impetus based on any belief (or other form of 'extra-rational knowledge', such as intuition or any a priori anything) is to introduce arbitrariness into the formulation of the ethic. From the perspective of any other person all beliefs (etc.) of all people are completely arbitrary. Locke relied on a belief in equality (from the first of his Two Treatises) to limit "liberty" to the person and property (the interest he emphasized to the exclusion of any other) of any other person.

Locating the 'ethic of justice' in material existence--both its determiner and its referents--avoids arbitrariness in its formulation. For that we can thank Warren J. Samuels, who all but defined "social power" as the ability to effect choices. Effecting choices is something all human beings have no choice but to do. It is integral to being human. To respect the capacity to choose of other people is to recognize them as fellow human beings. That boils down to a handful of absolute prohibitions: no killing, harming, coercing, stealing, or manipulating (which includes lying, cheating, etc.) in effecting any choice. Anyone who is not violating any of those prohibitions in that process is acting justly enough.