Does Uniform Integrability of a product martingale $M$ imply integrability of $\sup|M_n|$?

Question

Let $(M_n)$ be a non-negative uniformly integrable product martingale - i.e. $M_n=\prod_{j=1}^{n}X_j$ for independent non-negative r.v. $X_n$. The problem asks if there exists a random variable such that $|M_n|\leq Y$ and $E|Y|<\infty$. I learned that this is false in general from For a martingale $X$ does uniform integrability imply integrability of $\sup |X_{n}|$?. However, I am struggling to see which way it goes in the case of product martingales, as counterexamples are difficult to find. I haven't made much progress - I thought the Kakutani dichotomy might be useful, but my attempts with it didn't lead to anywhere. Any help is appreciated!