A Question about Terraced Matrices

Suppose that M is a lower triangular infinite matrix with constant row segments given by the positive decreasing sequence a(n), n≥0. Further, suppose that M is a bounded operator on the Hilbert space of square summable sequences.

Question: Does there exist such a sequence a(n) with (n+1)a(n) strictly decreasing to a limit L > 0 such that the associated operator M is hyponormal (i.e., the self-commutator of M is a positive operator)?

Note 1: The more general analogue for factorable matrices has been settled affirmatively by the weighted mean matrix generated by the sequence of positive integers.

Note 2: It is known that there exist sequences a(n) with (n+1)a(n) strictly increasing to a limit L > 0 such that the associated operator M is hyponormal.