Consider a hollow right circular cone
having radius 20 and height 16.
Find the maximum possible area of the square that can lie inside the cone.
Any square lies on a plane. When that plane slices the cone, the intersection is an ellipse, call it E.
Now slice a second plane through the cone perpendicular to the cone's axis and containing the center of the ellipse. That plane intersects the cone in a circle, call it C.
The diameter of C is equal to the minor axis of E. The inscribed square in E is larger than the inscribed square in C. But in order to have a chance to be larger than the square inscribed in the base, the major axis of E must be longer than the diameter of the base.
But increasing the length of the major axis of E requires tilting the slicing plane increasingly steeper against the base, which has the effect of shrinking the minor axis.
So based on this handwaving argument I conjecture that the largest square is the square inscribed in the base, regardless of the values chosen for the radius and height.