Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that $$\sum_{i,j}(G^5)_{ij}\le\sum_{i,j}(G^3)_{ij}\,?$$

This is true if it is additionally assumed that all off-diagonal entries of $G$ are the same.