Let $\mathbb{F}=(\mathcal{F_n})_{n \in \mathbb{N}}$ be a filtration on some probability space, $(X_n)$ a series of $\mathbb{F}$-adapated i.i.d RVs with mean $\mu$, and $(\tau_n)$ a series of stopping times with respect to $\mathbb{F}$ such that $\tau_n <\tau_{n+1}$ for all $n\in \mathbb{N}$. Under these circumstances we know that $\mathcal{F}_{\tau_n} \subseteq \mathcal{F}_{\tau_{n+1}}$, where $X_{\tau_n}$ is $\mathcal{F}_{\tau_n}$-measurable.
Since the original variables are i.i.d, is it possible to say without further assumptions that $\mathbb{E}\left[X_{\tau_{n+1}}| \mathcal{F}_{\tau_{n}}\right]=\mu$?
Thank you.
Edit: Actually it is possible to assume that $\mathcal{F}_n=\sigma\left(X_1,...,X_n\right)$.
Let $\mathcal{F}_n = \sigma(X_1,\dots,X_n)$ for all $n$. Let $X_n$ be a sequence of i.i.d. Bernoulli random variables with success probability $0.5$. Define the stopping times as follows: $$\tau_n = \inf\{k: \sum_{i=1}^k X_i = n\}.$$ Then $\tau_n$ is a strictly increasing sequence of a.s. finite $\mathcal{F}$ stopping times. However, note that $X_{\tau_n} = 1$ a.s. for any $n$. Thus, $$\mathbb{E}[X_{\tau_{n+1}}|\mathcal{F}_{\tau_n}] = 1 \neq 0.5 = \mu.$$