This is Sota Asai.
I have the following problem related to my minicourse.
Let $K$ be an algebraically closed field.
Consider the path algebra $A:=KQ$, where the quiver $Q$ has three vertices $1,2,3$ and three arrows $\alpha \colon 1 \to 2$, $\beta \colon 1 \to 2$ and $\gamma \colon 2 \to 3$.
Then, describe the non-rigid region $\mathsf{NR}$ and the purely non-rigid region $R_0$ of the real Grothendieck group $K_0(\operatorname{proj} A)_\mathbb{R}$ as explicitly as possible.
This path algebra appears in Example 5.8 of my paper "The wall-chamber structures of the real Grothendieck groups", where the wall-chamber structure on $K_0(\operatorname{proj} A)_\mathbb{R}$ is partially depicted.
The definitions of the non-rigid region and the purely non-rigid region are written in Section 6 of our preprint "Semistable torsion classes and canonical decompositions" with Osamu Iyama.
I will talk about them in the last lecture of my minicourse if time permits.
I am sorry for that there is insufficient information in this post.
I will add some useful references later.
Since this quiver has exactly 3 vertices, let's try to do the following things.
Determine the g-vectors of the indecomposable 2-term presilting complexes. The indecomposable 2-term presilting complexes except $P_i[1]$ bijectively correspond to the indecomposable rigid modules, or equivalently, the real Schur roots.
For each indecomposable 2-term presilting complex $U$, determine the open neighborhood $N_U$ of $C^+(U)$ and the local wall-chamber structure in $N_U$.
And I think that the following papers by Derksen-Weyman are useful.
The combinatorics of quiver representations
On the canonical decomposition of quiver representations