The factorization property of $ell^infty(X_k)$

In this paper we consider the following problem: Let $X_k$, be a Banach space with a normalized basis $(e_{(k,j)})_j$, whose biorthogonals are denoted by $(e_{(k,j)}^*)_j$, for $kinmathbb{N}$, let $Z=ell^infty(X_k:kinmathbb{N})$ be their $ell^infty$-sum, and let $T:Zto Z$ be a bounded linear operator, with a large diagonal, i.e. $$inf_{k,j} big|e^*_{(k,j)}(T(e_{(k,j)})big|>0.$$ Under which condition does the identity on $Z$ factor through $T$? The purpose of this paper is to formulate general conditions for which the answer is positive.