Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual conditions, a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $\mathbb {E}\left[ \bar X \right ]< \infty$ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its Snell envelope

$$ \hat{X}_\theta = \underset{\tau \in \mathcal T_{\theta,T} } {\text{ess sup}} \ \mathbb {E} \left[ X_\tau |\mathcal F_\theta \right] $$

where $\mathcal T_{\theta, T} := \{ \tau \quad \mathbb F -\text{stopping time}: \tau \in [0, T] \quad \text{and} \quad \tau \geq \theta \quad \mathbb P -\text{a.s.} \}$ and $T \in \mathbb R_+$ is a deterministic constant.

I'd like to understand how justify the following inequality:

$$\mathbb E \left [ \sup_{0\leq t\leq T} \hat X_t \right] \leq \mathbb E \left [ \sup_{0\leq t\leq T} \bar X_t \right] $$

where $\bar X_t = \mathbb E \left [ \bar X | \mathcal F_t\right]$

**Suplementary question**
Justify the following inequality:
$$\mathbb E \left [ \sup_{0\leq t\leq T} \hat X_t^p \right] \leq \mathbb E \left [ \sup_{0\leq t\leq T} \mathbb E \left [ \bar X ^p| \mathcal F_t\right] ^p\right] $$