I think the main concern for anyone considered the "strong objectivity" advocated in Harding's paper would be the emphasis on the importance of marginalized perspectives in the pursuit of knowledge: how can we make any claims of objective knowledge if it is necessary that the perspective be that of specific individuals. To reformulate the question: if knowledge is objective (this meaning that it is not bound to any single individual or perspective) how can we simultaneously maintain that certain perspectives are more apt to be able to gain such a knowledge.
The best reply to such a concern would be to restate the origin of such a perspective (the European Enlightenment tradition) and in doing so emphasize the prejudice of such a view: the methods of the European epistemological tradition have maintained that certain perspectives are more apt to be able to gain knowledge (in this case, those who are of higher status and are thus more educated in the methods of knowledge production) and are more able to achieve an objective perspective. The key difference between "strong objectivity" and the empiricist objective enterprise lies in the lack of acknowledgment on the side of the empiricist that there is in fact a bias which is being enshrined in the questions and hypotheses being considered. The empiricist lies in uncontested social territory: they have absorbed the traditions and the biases of the society of which they are a part and replicate them in a way which mirrors the society at large and renders them nearly unrecognizable. The "strong objectivist", conversely, lies at the margins of society at large and thus are in a key position to be able to see what is invisible to others, to criticize what is taken as a given and thus acheive a greater degree of objectivity than other methods which stress objectivity.
While this argument is convincing, it also seems that there must be a counter-act of critique to fully strength of the marginal account: the individual who makes the account may be able to see the biases hidden in the society at large, but may harbor their own biases which may not be accounted for. Further, such an actor may have absorbed certain biases which infect the rest of the society; the actors are marginalized, not outside of the society that they critique, for they must have some working familiarity of the system. Such an example is historically found in the lack of attention paid to women in the advancement of civil rights. If we consider such a difficulty, we run the risk of infinite regress, insamuch as there seems to be no position of which we can say with complete security that it is better able to engage in critique. This, of course, assuming that we value objectivity as an ultimate end; here we must take it upon ourselves to consider the origins of the value of objectivity yet again.