Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n}$ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where $\mathbf{1} \in \mathbb{R}^n$ is the vector of all $1$s. Is this true that spectral radius $\rho(P-P\mathbf{1}w^T) < 1$?

I already proved that it is true when $P$ is doubly stochastic. When $P$ is column stochastic, randomly generated examples in Matlab seem to confirm the statement. Any help would be appreciated.