The eighth Busemann-Petty problem asks the following question: If for an origin-symmetric convex body K⊂Rn, n≥3, we have fK(θ)=C(voln−1(K∩θ⊥))n+1∀θ∈Sn−1, where the constant C is independent of θ, must K be an ellipsoid? Here, fK is the is the curvature function (the reciprocal of the Gaussian curvature). We will show that the answer is affirmative for K close enough to the Euclidean ball in the Banach-Mazur distance.