= Solution
A stopping time $T$ is a <strong stationary time> if
$$
\mathbb P_x(X_T=y,T=t)=\pi(y)\mathbb P_x(T=t)
$$
for every $x,y,t$. Thus $X_T\sim\pi$ and is independent of $T$. The <separation distance> is
$$
s(t)=\max_{x,y}\left\{1-\frac{P^t(x,y)}{\pi(y)}\right\}.
$$
For any strong stationary time,
$$
\begin{aligned}
P^t(x,y)
&\geq\mathbb P_x(X_t=y,T\leq t)\\
&=\sum_{j\leq t}\sum_z
\mathbb P_x(T=j,X_j=z)P^{t-j}(z,y)\\
&=\pi(y)\mathbb P_x(T\leq t).
\end{aligned}
$$
Hence $1-P^t(x,y)/\pi(y)\leq\mathbb P_x(T>t)$, and maximizing proves the claim.
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