Nobody said anything about landuse=natural_reserve which is a strongly discouraged tag.  I am asserting that leisure=nature_reserve is for areas where "people go for leisure, to the nature reserve" (where they might hike, if they stick to the trail, or stay at the wildlife viewing area, with their binoculars, if they don't want to venture too far into the reserve.

While "special opportunities" do exist for people to visit the actual-wilderness or near-wilderness areas (like it appears this is) you must get a special permit, someone doing specific academic research can be seen to get clear to do botanical or biological or whatever studies, but the general public (who "leisure"), no.

leisure=nature_reserve is an older tag and newer tags (in the protect_class numbered or protection_class plain-English vein) are preferred, as they are much less ambiguous.  I really do think this might be a leisure=nature_reserve for some purposes, but both of us don't seem to know that (yet).

Try a site visit and see how difficult that is.  That should tell you if this is a (casual) "leisure" area.  I think the tagging is otherwise correct without this tag.