Isomorphism conjecture for generalized and twisted multifold extensions

Let $A$ be a piecewise hereditary algebra of tree type, let $\hat{A}$ be its repetitive category, and let $\phi$ be an automorphism of $\hat{A}$ with nonzero jump $n$. Write

$$\Lambda=\hat{A}/\langle\phi\rangle$$

for the generalized multifold extension, and define

$$\phi_0=(1\kern-.25em{\text{\rm l}}^{[0]})^{-1}\phi\nu_A^{-n}1\kern-.25em{\text{\rm l}}^{[0]}.$$

Then $T^n_{\phi_0}(A)$ denotes the corresponding twisted multifold extension. Isomorphism conjecture. The algebras $\hat{A}/\langle\phi\rangle$ and $T^n_{\phi_0}(A)$ are isomorphic. The conjecture strengthens the established derived-equivalence result for generalized multifold extensions of piecewise hereditary algebras of tree type; the source reports that it arose from examples in which the two algebras appeared not only derived equivalent but also isomorphic.