Martingale convergence in $L^\infty$

Question

Let $(\Omega,(\mathcal{F}_n)_n,\Bbb{P})$ be a filtered probability space and let $(X_n)_n$ be a discrete time martingale which is bounded in $L^\infty$, say $ \|X_n\|_\infty\leq 1 $ for all $n\in\Bbb{N}$. Martingale convergence theorems apply and show that $X_n$ converges almost surely to some (bounded) random variable $X_\infty$, and $X_n$ converges to $X_\infty$ in $L^p$ for any $p\in[1,+\infty)$: $$ X_n\xrightarrow{~\Bbb{P}\text{ a.s.}~}X_\infty,~ X_\infty \text{ is essentially bounded and } \forall p\in[1,+\infty),~ X_n\xrightarrow{~L^p~}X_\infty $$ This answer shows that convergence need not happen in $L^\infty$. Are there general results that imply $L^\infty$ convergence?