It hasn't been debunked because there is a wage gap, according to the first three articles you linked. You can argue about what constitutes "significant," but the first article notes that according to the BLS, "the gap is 19 cents when looking at weekly wages" and "even smaller, 14 cents, when you look at hourly wages." The AAUW report, which looks specifically at college graduates working full-time, similarly found a gap of 14-19 cents. Also, the first three articles you linked are the meaningful ones here; links 4 and 5 are about gender differences in salary negotiation, and the last is an opinion column. It and the CONSAD report are the only ones of your link set that claim the gender gap is "very insignificant."

The variable CONSAD is controlling for, at least from my reading, is "people who have interrupted their careers by temporarily withdrawing from the labor force or switching from full-time to part-time work" -- mostly women who take maternity leave or part-time jobs to raise children. The amount of money those women would make if they hadn't interrupted their career is an interesting data point to determine, but I would argue that by using that "adjusted" amount, CONSAD has moved the goalposts, because that's not the median gender wage gap as actually experienced. The various studies linked to by the WaPo and the AAUW study all seem to be in general agreement around the 14-19 cents range. Whether that's "significant" is subjective, I suppose, but it doesn't strike me as a small amount.