Homomorphisms between pure mapping class groups

Let $S$ and $S'$ be orientable finite-type surfaces of genus $g\geq 4$ and $g'$, respectively. We prove that every multitwist-preserving map between pure mapping class groups $\text{PMap}(S)\to \text{PMap}(S')$ is induced by a multi-embedding. As an application, we classify all homomorphisms $\text{PMap}(S)\to \text{PMap}(S')$ for $g\geq 4$ and $g' \leq 6\cdot 2^{g-4}$.