Which mountain huts might be open outside of Mt. Fuji's regular season?

I'm considering climbing Mt. Fuji in mid-June, slightly before the regular season starts (regular season is July to mid-September). That way, it might be less crowded and a little more peaceful.

I'm not worried about the climb itself personally, as I am an experienced mountaineer and can judge conditions well enough to determine whether to continue or turn back.

However, most climbing schedules include one night in a mountain hut part of the way up, where you get dinner and sleep before going up to the summit. Is there an easy way to find out which huts might be open earlier than the official start?

Unfortunately, carrying sleeping & cooking gear is just not logistically possible for me on this trip (though it would definitely resolve this issue, and probably even be more fun).