Lower bound on the quadratic form of a diagonally dominant stochastic matrix?

Question

Let $p$ be a generic (column) probability vector with $n$ entries, and $M$ be a diagonally dominant stochastic matrix with $n$ rows and colums, not necessarily symmetric.

Is it always true that $p^T M p \geq \frac{1}{n}$ (as is the case when $M$ is the identity)?