  Consider a tensor product of simple dyadic shifts defined below. We prove
here that for dyadic bi-parameter repeated commutator its norm can be estimated
from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems
in Section 8 at the end of this article. But this is done below under an extra
assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully
analyze what goes wrong in the absence of this extra assumption. At the end of
this note we also list a counterexample to the existing proof of
characterization of bi-parameter repeated commutator with the Hilbert
transforms. This is a counterexample to the proof, and it is not a
counterexample to the statement of factorization result in bi-disc, or to
Nehari's theorem in bi-disc. To the best of our knowledge Nehari's theorem on
bi-disc is still open. Moreover its dyadic bi-parameter version considered in
the present paper is also still open for general symbol without any extra
restrictions.
