  For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say
that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We
say that $A$ is Schurian-finite if there are only finitely many isomorphism
classes of Schurian $A$-modules, and Schurian-infinite otherwise. In this
paper, we build on the work of Ariki and the second author to show that all
blocks of type $A$ Hecke algebras of weight at least $2$ in quantum
characteristic $e \geq 3$ are Schurian-infinite. This proves that if $e \geq 3$
then blocks of type $A$ Hecke algebras are Schurian-finite if and only if they
are representation-finite.
