Is the ban on women in Mt. Omine legally enforced?

Mount Omine in Japan is said to be off-limits to women. However, it isn't clear (at least from a quick browse of English-language websites) whether the ban is legally enforced, or whether the mountain is on private property. On google maps the mountain and the surrounding area is shaded in green, which usually indicates that it is some sort of national/state park, i.e. government-owned property. If that's the case, I find it hard to believe that there would be a police-enforced ban on women, in a developed first-world country, in a place otherwise open to the public. Does anyone have definitive information about the rules on Mt. Omine?

Note: Please do not answer with something like "It may be legally OK, but it would be deeply disrespectful, and you should respect their culture ...", etc. I understand that this question may be controversial, but my intention is not to start a debate on this topic.