Let $\Lambda$ be a finite-dimensional algebra. The classical Auslander-Reiten translation $\tau$ is defined as $\tau= \mathrm{D}\mathrm{Tr}$ where $\mathrm{Tr}$ is a duality between $\underline{\mathrm{mod}}\;\Lambda$ and $\underline{\mathrm{mod}}\;\Lambda^{\mathrm{op}}$ and $\mathrm{D}$ is a duality between $\mathrm{mod}\;\Lambda$ and $\mathrm{mod}\;\Lambda^{\mathrm{op}}$. As dualities flip monomorphisms to epimorphisms and vice versa, it follows that $\tau$ preserves monomorphisms and epimorphisms.
In higher homological algebra one makes use of the higher Auslander-Reiten translation $\tau_n=\tau\Omega^{n-1}$, where $\Omega$ denotes the syzygy of a module (i.e. the kernel of a projective cover). If $\mathcal{C}$ is an $n$-cluster tilting subcategory of $\mathrm{mod}\;\Lambda$, and $f$ is an epimorphism between modules in $\mathcal{C}$, then in examples I can see that $\tau_n(f)$ is an epimorphism if $n$ is odd and is a monomorphism if $n$ is even (and the dual statement holds if $f$ is a monomorphism). Is this known to be always true? If not, is there a counterexample?