If it does, then the goal of that kind of equality is a false goal.
If not, then you’re right.
Right now, there isn’t really choice, because few countries have had equal leave rights, and those that do have had them for such a relatively short amount of time that we have no idea. So we haven’t been able to test any sort of hypothesis at all. All we have is culturally bound gut feelings, but if we just listened to those then we would still think that there are only four elements, that the earth is flat, etc.
And in the US we don’t even have legally mandated paid maternity leave, so even that would be a start.
In this case, we are asking whether two hypothetical people who are completely identical in every way except for their genitals will receive the same opportunities, rewards, and treatment from their employers and their peers.
It is not for public policy to speculate on which parent will produce more value for the baby or the household by staying at home. The point you were responding to originally was this: If you mandate more parental leave for women than for men, you end up creating an incentive for employers to hire men instead of women (or choose to let women go when layoffs come around), because their benefits cost less.
That is the type of gender inequality we are talking about, and I hope you will agree that fixing it is not a "false goal".
Shinjirō Koizumi caused controversy when he took two weeks of his family leave as a minister because Japanese men never take it due to cultural reasons even though they are legally entitled to a full year of paternal leave. https://www.cnn.com/2020/01/16/asia/japan-koizumi-paternity-...
And he was the son of a former prime minister, so even he was not powerful enough to be immune from criticism of this kind