Let K be an origin-symmetric convex body in dimension 3 or higher. For each unit vector u, consider the cone whose base is the projection of K on the hyperplane perpendicular to u, and whose height is the value of the radial function of K in the direction u. If the volume of this cone is constant independent of u, must K be an ellipsoid? This question is dual to the statement of Busemann-Petty Problem 5. We obtain an affirmative result for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance. The proof is more involved that the local affirmative answer to Busemann-Petty Problem 5. This is joint work with F. Nazarov, D. Ryabogin and V. Yaskin.