Weakly Diagonally Dominant with Positive Diagonals

Question

Suppose $A \in \mathbb{R}^{n\times n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of complex eigenvalues, have non-negative real parts)? Note that this holds true for strictly diagonally dominant matrices: if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.