On isomorphisms of generalized multifold extensions of algebras without nonzero oriented cycles

Assume that a basic algebra $A$ over an algebraically closed field $\Bbbk$ with a basic set $A_0$ of primitive idempotents has the property that $eAe=\Bbbk$ for all $e \in A_0$. Let $n$ be a nonzero integer, and $\phi$ and $\psi$ two automorphisms of the repetitive category $\hat{A}$ of $A$ with jump $n$ (namely, they send $A^{[0]}$ to $A^{[n]}$, where $A^{[i]}$ is the $i$-th copy of $A$ in $\hat{A}$ for all $i \in \mathbb{Z}$). If $\phi$ and $\psi$ coincide on the objects and if there exists a map $\rho \colon A_0 \to \Bbbk$ such that $\rho_0(y)\phi_0(a)=\psi_0(a)\rho _0(x)$ for all morphisms $a\in A(x,y)$, then the orbit categories $\hat{A}/\langle \phi \rangle$ and $\hat{A}/\langle \psi \rangle$ are isomorphic as $\mathbb{Z}$-graded categories.