To What Class of Fractals Does the Alexander-Orbach Conjecture Apply?

Alexander and Orbach have recently made the remarkable numerical discovery that for the incipient infinite cluster in percolation the ratio of $d_f$ (the fractal dimension of the aggregate) to $d_w$ (the fractal dimension of a random walk on the aggregate) is approximately "superuniversal"—independent of $d$ for $d>\mathrm{}1\mathrm{}$. Does this discovery also hold for aggregates other than percolation? A plausibility argument (rigorous for the Cayley tree) is presented that it should hold, exactly, for "homogeneous" fractals, but need not for nonhomogeneous fractals such as the percolation backbone, the Sierpinski gasket, and the Havlin carpet.