Suppose that $X=(X_{n})_{n\geq 1}$ and $Y=(Y_{n})_{n\geq 1}$ are two independent homogeneous Markov chains on the state space $S=\{0,1\}$, with transition probability matrices $P_{1}$ and $P_{2}$ given by
$$P_{1}=\begin{bmatrix} 1-\theta&\theta\\\theta &1-\theta \end{bmatrix}$$
$$P_{2}=\begin{bmatrix} \theta&1-\theta\\1-\theta &\theta \end{bmatrix},$$
where $\theta\in [0,1]$ is a known constant.
Suppose $T$ is an almost surely finite stopping time with respect to the filtration generated by the $X$ process. Then, by the strong Markov property, we know that for any fixed constant $\tau>0$,
$$ P_{1}(X_{T}=x'|X_{T-\tau}=x)=P_{1}(X_{\tau}=x'|X_{0}=x) $$ for all $x,x'\in S$.
However, I am not quite sure if the same property is true with $P_{2}$ as well. In other words, I am not sure of the validity of the above equation with $P_{1}$ replaced by $P_{2}$.
Can anyone kindly let me know how I can evaluate $$ P_{2}(X_{T}=x'|X_{T-\tau}=x)$$ for all $x,x'\in S$?
Any hints or suggestions are most welcome.