They bring Alex and a moldy apple to the cave, and none of them go. Some further experimentation reveals that the problem with the moldy apple is both the seeds and the mold - they don't have the wherewithal to really control for the presence of spores in an environment without grown mold, or the presence and evolution of bacteria when crossing the threshold.
From a legal perspective, they don't have a lot of ground to stand on. The property is the government's, and it isn't selling. Laws tangential to the issue, such as those relating to oil or caves, could get it to be argued several ways, and the only way to make it work as they want to is if they actually create another portal like that on land they actually own. Or it would be, if it weren't for the more relevant Outer Space Treaty, which most relevantly says that celestial resources are the common heritage of mankind, and must not be explored by any one single State.
And Isabella's week off ends and they have to return to New York (or, in Sadde's case, go there for the first time).