Given a Banach space X with an unconditional basis, we consider
the following question: does the identity on X factor through every bounded
operator on X with large diagonal relative to the unconditional basis? We show
that on Gowers? space with its unconditional basis there exists an operator for
which the answer to the question is negative. By contrast, for any operator
on the mixed-norm Hardy spaces Hp(Hq), where 1 p, q < 1, with the
bi-parameter Haar system, this problem always has a positive solution. The
one-parameter Hp spaces were treated first by Andrew  in 1979.