Let $A=KQ$ be a path algebra over a field $K$ with finite connected quiver $Q$.
A slope function $\mu$ is a function of the form $\mu=\sigma/dim$ defined on the Grothendieck group $K\_0(A) \setminus 0$ (without the zero module), where $\sigma$ is linear and dim is just the sum of entries or equivalently the dimension on modules. A module $M$ of $KQ$ is called $\mu$-stable if $\mu(N) < \mu(M)$ for every non-zero submodule $N$ of $M$.


Note that a slope function $\mu$ on a dimension vector $[a\_1,...,a\_n]$ is simply a function given by $\mu([a\_1,...,a\_n])=x\_1 a\_1 + ... x\_n a\_n/(a\_1+...+a\_n)$ for some $x\_1,...,x\_n$.


Reineke's conjecture (see conjecture 7.1 in <https://link.springer.com/article/10.1007/s00222-002-0273-4>) can be stated as follows:



> 
> On every Dynkin type quiver $Q$ there exists a slope function $\mu$ having the property that every indecomposable module is $\mu$-stable.
> 
> 
> 


I made a program that verified this conjecture for all Dynkin quiver with at most 6 vertices.
But surprsisingly, it seems it is wrong for Dynkin type $Q=E\_7$ with the following orientation (or I have a stupid thinking error which is very likely since I started with this only 2 weeks ago):



```
Quiver( ["1","2","3","4","5","6","7"], [["2","1","a_1"],["3","2","a_2"],["4","3","a_3"],["5","4","a_4"],["6","5","a_5"],["3","7","a_6"]] )

```

Fix this $Q$ for the rest now and assume a slope function $\mu$ given by the $x\_i$ exists.


Now we have the following five indecomposable $KQ$-modules $M\_i$ with submodules $N\_i$ for $i=1,...,5$ (note that indecomposable modules are uniquely determined by their dimension vectors). The inclusions are in fact irreducible maps:


1. $[0,0,1,1,1,0,1]-> [1,1,2,2,2,1,1]$


Applying $\mu$ leads to the inequality $0<4x\_1+4x\_2-2x\_3-2x\_4-2x\_5+4x\_6-6x\_7$.


2. $[0,1,1,1,0,0,0] -> [0,1,2,2,1,0,1]$


Applying $\mu$ leads to the inequality $0<-4x\_2-x\_3-x\_4+3x\_5+3x\_7$.


3. $[0,0,1,0,0,0,0] ->[0,0,1,1,0,0,0]$


Applying $\mu$ leads to the inequality $0<-x\_3+x\_4$.


4. $[0,1,1,0,0,0,0] -> [0,1,2,1,0,0,1]$


Applying $\mu$ leads to the inequality $0<-3x\_2-x\_3+2x\_4+2x\_7$.


5. $[1,1,1,1,1,1,1] -> [1,2,2,1,1,1,1]$


Applying $\mu$ leads to the inequality $0 < -2x\_1 +5x\_2 + 5x\_3 - 2x\_4 -2x\_5-2x\_6-2x\_7$.


Here a proof by hand that the system of those five inequalities has no solution:


(1)$0<4x\_1+4x\_2-2x\_3-2x\_4-2x\_5+4x\_6-6x\_7$


(2)$0<-4x\_2-x\_3-x\_4+3x\_5+3x\_7$


(3)$0<-x\_3+x\_4$


(4)$0<-3x\_2-x\_3+2x\_4+2x\_7$


(5)$0 < -2x\_1 +5x\_2 + 5x\_3 - 2x\_4 -2x\_5-2x\_6-2x\_7$


adding 1/2 \*(1) to (5) gives the inequalities:


(1)$0<4x\_1+4x\_2-2x\_3-2x\_4-2x\_5+4x\_6-6x\_7$


(2)$0<-4x\_2-x\_3-x\_4+3x\_5+3x\_7$


(3)$0<-x\_3+x\_4$


(4)$0<-3x\_2-x\_3+2x\_4+2x\_7$


(5)$0 < 7x\_2 + 4x\_3 - 3x\_4 -3x\_5-5x\_7$


adding now (2) and (4) to (5) gives:


(1)$0<4x\_1+4x\_2-2x\_3-2x\_4-2x\_5+4x\_6-6x\_7$


(2)$0<-4x\_2-x\_3-x\_4+3x\_5+3x\_7$


(3)$0<-x\_3+x\_4$


(4)$0<-3x\_2-x\_3+2x\_4+2x\_7$


(5)$0<2x\_3-2x\_4$


But now clearly inequalities (3) and (5) give a contradiction.


Thus since there is no solution, no slope function can exist for this $Q$ of Dynkin type $E\_7$.



Is this a counterexample to Reineke's conjecture on total stability conditions for Dynkin type quivers?
Your argument looks correct to me. Note that the corresponding question for the derived category has been answered positively, and a parameterisation of total stability conditions given by [QiuYu and ZhangXiaoting](https://arxiv.org/abs/2202.00092). In the case of $\mathsf{E}\_7$, this space is $7$-dimensional (over $\mathbb{C})$.


