fwiw: previous editions (on which 5e's version is based) said: "Half the sphere projects above the ground, and the lower hemisphere passes through the ground" even though "The interior of the hut is a hemisphere".
5e's unfortunate rewording created ambiguity, but I do not see an indication...
Be your players' biggest cheerleader. D&D is is all about fantasy and wish fulfillment. Commend him. If nothing else he evidently put a lot of time and effort crafting his character, which is the sort of engagement to strive for in players. Let him have that win. Then marvel at something the...