Decay estimates connected with Toeplitzness of composition operators

In a higher dimensional version of an earlier conjecture of Nazarov and Shapiro, the truth of which would imply that any composition operator on the second Hardy space is weakly asymptotically Toeplitz, Shayya proved that the arithmetic means of the main diagonal entries of the matrix of coefficients connected with the conjecture, decay like 1/log N. But the proof does not extend to other diagonal entries. We prove that the arithmetic means of the entries along each diagonal decay like 1/(log N loglog N) uniformly.