Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus r}$. Suppose also that, for every vertex $v$ of $Q$, we have $\dim F_v \leq \dim E_v$.

Does it follow that $F$ injects into $E$?

There are results like this in the work of Derksen, Schofield and Weyman, but I can't find this particular statement. Thanks!