DWZ-mutation homeomorphism conjecture for Ziegler spectra

Let $(Q,S)$ be a quiver with potential, let $Q_0$ be its set of vertices, and let $k\in Q_0$ be a vertex not incident to any $2$-cycle of $Q$. Write $\mathcal{P}(Q,S)$ for the Jacobian algebra of $(Q,S)$, and let $\mu_k(Q,S)$ denote its DWZ-mutation at $k$.

DWZ-mutation homeomorphism conjecture. The DWZ-mutation of modules is a homeomorphism between the Ziegler spectra of the Jacobian algebras

$$\mathcal{P}(Q,S)\quad\text{and}\quad\mathcal{P}(\mu_k(Q,S)).$$

The conjecture proposes that the homeomorphism between Ziegler spectra proved for quivers associated with triangulations of unpunctured annuli extends to every quiver with potential and every vertex at which DWZ mutation is defined without a $2$-cycle incidence. Its proof is deferred to sequels to the present paper.