Consider a $\pi$-stationary finite state Markov transition matrix $P$. Sample an $n$-length sequence from the Markov process sampling the initial state from $\pi$, and call this $S^n = \{ S_1,\cdots,S_n \}$. For each $i = 1,\cdots,n$ generate an independent $Z_i \sim \textsf{Geo} (1/2)$ random variable. Generate a new sequence $S'$ by replicating each state $S_i$ consecutively $Z_i$ times. That is, $$S' = \{ \underbrace{S_1,\cdots,S_1}_{Z_1}, \underbrace{S_2,\cdots,S_2}_{Z_2},\cdots,\underbrace{S_n,\cdots,S_n}_{Z_n}\}.$$
In addition, generate a sequence $S''$ by sampling a sequence of length $n' = \sum_{i=1}^n Z_i$ from the lazy Markov process with transition matrix $\frac12 (P + I)$ (where the initial state is again sampled from $\pi$ which is the stationary distribution of this Markov process)
My question is: $S'' \overset{d}{=} S'$?