Uniform integrability of continuous function of conditional expectations

Question

• Let $X$ be an integrable real valued random variable.
• Let $\sigma_n$ be a sub-sigma-algebra such that $\sigma(X) = \sigma(\cup_{n\in\mathbb{N}} \sigma_n)$.
• Suppose $f(X)$ is integrable where function $f:\mathbb{R}\to\mathbb{R}$ is continuous.
• Suppose $f(E[X \mid \sigma_n])$ is integrable for all $n$.

I know that $\{E[X \mid \sigma_n]:n\in\mathbb{N}\}$ is uniformly integrable.

My question is whether $\{f(E[X \mid \sigma_n]):n\in\mathbb{N}\}$ is also uniformly integrable?

Thank you.