What are the requirements for a diagonally dominant matrix to have a diagonally dominant square?

Question

Given a diagonally dominant matrix, i.e., a matrix $A$ whose entries satisfy $$|a_{ii}| \geq \sum_{j\neq i}|a_{ij}|\quad\forall\enspace i,$$ what other constraints are required to guarantee that $A^2$ is also diagonally dominant?