Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced by $\sigma$ and $\{p_{\sigma}\}$ is a distribution. Suppose $v$ is an $\epsilon$-approximate stationary distribution of $M$, i.e., $$\|v-Mv\|_{TV}\leq\epsilon,$$ where $\|\cdot \|_{TV}$ is the total variance. Does it imply the following $$\sum_{\sigma}p_{\sigma}\|v-M_{\sigma} v\|_{TV}\leq \epsilon',$$ where $\epsilon'=f(\epsilon)$ only depends on $\epsilon$, and goes to 0 if $\epsilon$ goes to 0. In other words, is $v$ also an approximate stationary distribution of $M_{\sigma}$ in expectation? It is not hard to prove when $\epsilon=0$. I wonder whether this problem has been studied.

Thanks.