Definitions.
Let $X_0, X_1, X_2, \ldots$ be a Markov chain on a finite state space $\Omega$. Assume the Markov chain is irreducible and aperiodic. So there is a unique stationary distribution $\pi$.
An $\mathbb N$ valued random variable $T$ is called a stopping time if the event $T=t$ depends only on $X_0, \ldots, X_t$ (I guess this formally means that $T=t$ is measurable in the $\sigma$-algebra generated by $X_0, \ldots, X_t$).
A stopping time $T$ is called a strong stationary time (SST) if $$P[X_t=x|T=t]= \pi(x)$$ for all $x\in \Omega$.
Let $p^{(t)}_x(y)$ denote the probability that $X_t=y$ if the chain starts at $x$, that is, if $P[X_0=x]=1$. We write $\Delta_x(t)$ to denote the total variation distance between $p^{(t)}_x$ and $\pi$.
Questions.
Let $T$ be a strong stationary time. In this document, Claim 4.2 states that $$\Delta_x(t)\leq P[T>t|\ X_0=x]$$ and the author says that this should be intuitively reasonable.
I am quite stumped. Can somebody elaborate on as to what is the intuitive reason to expect this. In fact, I do not have much in the way of intuition regarding the very concept of SST.
Next, in the proof of Claim 4.2 on Pg 4, a line reads
"... we denote by $T_x$ the SST for the chain started at state $x$..."
I am not sure what is the meaning of this. Does this mean that for each $x$ there is associated a (unique) SST?