TITLE: Is this a counterexample to Reineke's conjecture on total stability conditions for Dynkin type quivers?
QUESTION [8 upvotes]: 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:

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

Applying $\mu$ leads to the inequality $0<4x_1+4x_2-2x_3-2x_4-2x_5+4x_6-6x_7$.

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

Applying $\mu$ leads to the inequality $0<-4x_2-x_3-x_4+3x_5+3x_7$.

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

Applying $\mu$ leads to the inequality $0<-x_3+x_4$.

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

Applying $\mu$ leads to the inequality $0<-3x_2-x_3+2x_4+2x_7$.

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

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 $E_7$.

REPLY [3 votes]: 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. In the case of $\mathsf{E}_7$, this space is $7$-dimensional (over $\mathbb{C})$.