Let $D$ be a diagonally dominant matrix (both row and column-wise). Assume $D$ is not necessarily symmetric.

Consider the following block representation of $D^{-1}$: \begin{equation} D^{-1}=\begin{bmatrix} B_1 & B_2 \\ B_3 & B_4 \end{bmatrix}. \end{equation} Here assume that $B_i$ are all square matrices.

My question is whether $C=(B_1^{-1} + B_1^{-T})$ is also diagonally dominant. Note that if $B_1=D^{-1}$, then we have $C=(D+D^T)$, and this trivially holds.