The hereditary abelian subcategory conjecture for the non-strong minimal right determiner example

Let $Q$ be the strongly locally finite quiver from the preceding example, let $M$ be its displayed representation, and let $P_{i'}$ denote the corresponding projective representations. Set $\mathcal C'_0$ to be the full subcategory of $\operatorname{rep}(Q)$ consisting of finite direct sums of $M$ and $P_{i'}$ for $i'\geq1$. Inductively, let $\mathcal C'_n$ be the full subcategory closed under direct summands and finite direct sums, containing the kernels and cokernels of morphisms in $\mathcal C'_{n-1}$ and every object $B$ occurring in an exact sequence

$$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$

with $A,C\in\mathcal C'_{n-1}$, and put

$$\mathcal C':=\bigcup_{n\geq0}\mathcal C'_n.$$

Hereditary abelian subcategory conjecture. The category $\mathcal C'$ is a $\operatorname{Hom}$-finite hereditary abelian category.