I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X_n)_{n\geq 1}$ are independent r.v.'s, also subexponential, then $\sum_{n=1}^\tau X_n$ aso has a subexponential distribution.

Specifically, I would like to know if the following statement is known:

Let $(X_n)_{n\geq 1}$ be independent random variables satisfying $\mathbb{E}[e^{X_n}] \leq 1$ for all $n$, and $\tau$ be a stopping time. Suppose $\mathbb{E}[e^{\alpha \tau}] \leq e^\beta$ for some $\alpha >0$ and $\beta<\infty$. Then $\mathbb{E}[e^{\sum_{n=1}^\tau X_n}] \leq 1$.