Factorization of the identity through operators with large diagonal

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 $H^p(H^q)$, where $1 \leq p,q < \infty$, with the bi-parameter Haar system, this problem always has a positive solution. The one-parameter $H^p$ spaces were treated first by Andrew in 1979.