The discrete Gronwall's inequality states that if $x_n$ and and $u_n$ are non-negative sequences such that

$$ x_{n+1}\le a+\sum_{k=0}^n u_k x_k$$
then $$x_n\le a\prod_{k=0}^{n-1} (1+u_k)$$
(It can be shown using a simple induction as in https://math.stackexchange.com/questions/325565/gronwalls-lemma-discrete-version)

For the analysis of a random process, I came with an inequality that is similar to the one above but with a conditional expectation instead of an almost-sure inequality.

More precisely, let $X_n$ and $U_n$ be two discrete time stochastic processes adapted to a filtration $F_n$ such that $X_n>0$, $U_n\in(0,1)$, and $$ E[X_{n+1} | F_n] \le a - \sum_{k=0}^n U_k X_k $$ Let $T$ be a stopping time that is almost surely bounded. I am wondering if the following inequality holds: $$E[X_T]\le a E[\prod_{k=0}^{T-1}(1-U_k)]$$

Is it a known result? I have the impression that the induction used in the deterministic case does not work. If it helps, I can assume that $U_k$ is $F_{k-1}$ measurable for all $k$.