Block $LU$ factorization is stable for block matrices whose inverses are block diagonally dominant

Let $A\in M_n(\mathbb C)$ and let its inverse $B=A^{-1}$ be represented as an $m\times m$ block matrix that is block diagonally dominant either by rows or by columns w.r.t. a certain matrix norm. We show that $A$ possesses a block $LU$ factorization w.r.t. the partitioning defined by $B$, and the growth factor for $A$ in this factorization is bounded above by $1+\sigma$,where $\sigma=\max_{1\le i\le m}\sigma_i$ and the $\sigma_i$, $0\le\sigma_i\le1$, are the row (column) block dominance factors of $B$. Further, the off-diagonal blocks of $A$ (and of its block Schur complements) satisfy the relations
$$ \|A_{ji}A_{ii}^{-1}\|\le\sigma_j, \qquad j\ne i. $$