Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras

Let $\mathbb{F}$ be a field, $q$ a non-zero element of $\mathbb{F}$ and $\mathcal{H}_{n}=\mathcal{H}_{\mathbb{F},q}(\mathfrak{S}_n)$ the Iwahori-Hecke algebra of the symmetric group $\mathfrak{S}_n$. If $B$ is a block of $\mathcal{H}_{n}$ of $e$-weight $3$ and the characteristic of $\mathbb{F}$ is at least $5$, we prove that the decomposition numbers for $B$ are all at most $1$. In particular, the decomposition numbers for a $p$-block of $\mathfrak{S}_n$ of defect $3$ are all at most $1$.