Sum of conditional expectations of a bounded stochastic process

Question

Is there a proof for the following statement or is there a counter-example?

Let $\{X_t\}$ be a stochastic process adapted to the filtration $\{\mathcal{F}_t\}$. Assuming $0 \leq X_t \leq 1$, and $\sum_{t=1}^{T} X_t \leq c$ almost surely for some fixed $c \in \mathbb{R}$. then it holds that: \begin{align} \sum_{t=1}^{T} \mathbb{E}[X_t | \mathcal{F}_{t-1}] \leq c. \end{align}

Answer 1

Here's a simple counterexample.

Let $X_1$ be uniform on $[0,1]$ and let $X_2=1-X_1$. Then we have bound (in fact, equality): $$X_1 + X_2 \leq c\equiv 1$$ but: $$E[X_2|X_1]+E[X_1] = 3/2-X_1$$ which exceeds $c\equiv 1$ on the event $\{0\leq X_1< 1/2\}$.