second largest singular value of a doubly-stochastic irreducible primitive matrix

Question

For an irreducible primitive stochastic matrix $A$, $1$ is the largest eigenvalue and is simple, then the second largest eigenvalue has module strictly smaller than $1$. I also know that the largest singular value of $A$ needs to be $1$ because $A$ is doubly stochastic. What about the second largest singular value of $A$? Can we claim that it is strictly smaller than $1$? Thanks in advance.