I am not sure if this would be possible for a non-ASB scenario unless the POD was early in the course of human evolution. The fact is that males are naturally bigger, stronger, faster, more intelligent, more creative, and more rational than females... on average. This is not to say that women do not have their fair share of accomplishment, but overall human males are better in most areas than human females.
In some species the gender gap is greater. At one extreme, in some species of angler fish the male metamorphoses into a wart-like growth on the female and does nothing but generate sperm. In angler fish, the female is superior and the male is virtually worthless. Some species have very little sexual dimorphism or almost none, and others have much greater sexual dimorphism. Human females are NOT worthless, but they are not quite as capable as males.
Unfortunately, human women are smaller and more frail than men, which disadvantages them in contests of strength or athletics, and generally more hormonal and emotional, which limits the capacity to think as logically as most men. There are of course tasks in which human females are far superior to human males. Most notably, tasks which require emotionality or nurturing ability. Women are naturally better than men at caring for children and animals, drama, fashion, and cleaning. Men are better than women at being warriors, statesmen, scientists, artists, and laborers.
Were the earliest societies male-dominated or female-dominated? The case can be made that prehistoric cultures were patriarchal, or matriarchal, or egalitarian. The problem is that when one attaches such labels to prehistoric cultures, one tends to project the current situation. In fact, prehistoric societies were neither patriarchal, matriarchal, or egalitarian, though they had aspects of the three.
Domestic life was matriarchal and female-dominated, as the home was almost entirely the domain of women. However, the women stayed behind at the camp or village, caring for the children or their families back home, keeping the huts, cave, or campsite clean, preparing food, and conducting the daily affairs of hearth, home, and village. The men would leave to go hunting, so subsistence activity was entirely patriarchal and male dominated. Hunter-gatherer societies have rites of passage between boyhood and manhood because male youths leave the "domain of women" and joins the men of the band/clan/tribe to hunt. Prehistoric life only SEEMS so egalitarian because there were natural gender roles to follow, allowing men and women to live separately. Feminist archaelogists may have interpreted fertility godesses such as the Venus of Willendorf and other figurines or wall paintings of pregnant women as Mother Goddess figures, while non-feminist archaeologists have had alternative explanations. Prehistoric humanity did not have notions of sexism or feminism and gender equality the way we think of these concepts. Rather prehistoric concepts were probably more along separate-but-equal lines, in which each gender had roles, and motherhood was a sacred role for females.
With the agricultural revolutions, society became sedentary. Men and women began spending more time together, many of the adolescent male rites of passage vanished or diminished, and society became more structured. It did not take long for males to establish dominance. The fertility goddesses of early agricultural societies of the Neolithic, Chalcolithic, and Early Bronze Age were very different in nature compared to their Upper Paleolithic and Mesolithic counterparts. Contrary to feminist archaeologists and feminist anthropologists, female deities did not always represent matriarchal societies. Rather, such often deities mythologized the conquest of nature by man. There was dual symbolism in these motiffs, the discovery of the power to "create life" (at first limited to childbirth among hunter-gatherers, later represented by plant cultivation and animal husbandry) by mankind, and a token of the power of man over woman.
Matriarchal and egalitarian civilizations are the exception rather than the rule. Most societies are male-dominated, some more than others. One ancient culture that was intensively male dominated was the proto-Indo-European culture of Anatolia, which colonized Europe and Central Asia in neolithic/chalcolithic times.
It was only natural that men became the dominant sex. Perhaps in about ten million years, a new sapient species will evolve to replace long-extinct humans, hopefully one with very little sexual dimorphism, and thus free of the inevitable problems stemming from conflict over gender roles such as sexism, misogyny (on one hand), misandry/feminism (on one hand), and homophobia.