Reineke's total stability conjecture for Dynkin quivers

Let $Q$ be a finite Dynkin quiver and let $A=KQ$ be its path algebra. A slope function is a map $\mu:K_0(A)\setminus\{0\}\to\mathbb{R}$ of the form $\mu=\frac{\theta}{\kappa}$, where $\theta$ and $\kappa$ are linear functions on $K_0(A)$ and $\kappa(M)>0$ for every nonzero $M\in K_0(A)$. A slope function defines a total stability condition if every indecomposable $A$-module is $\mu$-stable. Here $\dim$ denotes the function assigning to a module its dimension over the base field. Reineke's total stability conjecture. There exists a slope function of the form $\mu=\frac{\theta}{\dim}$ that defines a total stability condition for $A$. The conjecture is refuted: a path algebra of Dynkin type $E_7$ with a specific orientation admits no slope function of this form defining a total stability condition, providing a counterexample to the conjecture.