Let $Q$ be an acyclic quiver. Is it true that if $P$ is a preprojective representation of $Q$ and $r\geq 0$, there exists $s\geq r$ and a preprojective $P'$ with an injective morphism $$ P\rightarrow \tau^{-s}P' $$ where $\tau^-,\tau$ are the Auslander-Reiten translates?

In wild type, it follows frop Proposition 5.3 in this book, and in type $A_n^{(1)}$, it follows from this paper, Corollary 2.2.

I am not able to prove it for other affine quivers.