When feuding with strangers on Twitter about the gender wage gap, a common explanation for the much smaller gender wage gap at the bottom of the labour market is the power is the minimum wage to uplift women’s wages and with it narrow the gender wage gap at the bottom of the labour market.
Below I have plotted the minimum wage as a percentage of the median wage for the USA along with the gender wage gap for the bottom 10% of the U.S. labour market. Both are expressed as percentages of the median wage.
Source: OECD Employment Database and Data extracted on 30 Nov 2015 04:03 UTC (GMT) from OECD.Stat.
From what I can make of the above chart, the gender wage gap at the bottom of the labour market was closing rapidly when the minimum wage was falling as a percentage of the median wage. That gender wage gap then stabilised when the minimum wage stabilised as a percentage of the median. The gender wage gap increased when the minimum wage increased as a percentage of the median wage for full-time employees.
Should not this relationship between the minimum wage and the gender wage gap be the other way around if the minimum wage is a force for closing the gender wage gap at the bottom of the labour market?
Utopia, you are standing in it! 
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