On a proposed characterization of schatten-class composition operators

For an analytic function φ which maps the open unit disc D to itself, let C_φ be the operator of composition with φ on the Bergman space L_a²(D, dA). It has been a longstanding problem to determine whether or not the membership of C_φ in the Schatten class C_p, 1 < p < ∞, is equivalent to the condition that the function z → {(1 - |z|²)/(1 - |(z)|²)}^p has a finite integral with respect to the Möbius-invariant measure dλ(z) = (1 - |z|²)^-2dA(z) on D. We show that the answer is negative when 2 < p < ∞.