Conjecture on Auslander–Reiten components outside the finitely presented representations

Let $Q$ be a strongly locally finite quiver. Write \text{\mathit\Gamma}_{{\rm rep}(Q)} for the Auslander–Reiten quiver of ${\rm rep}(Q)$, and let $\overline{\mystrut(5,0) {\rm rep\hspace{-1pt}}}\,(Q)$ denote the subcategory of finitely presented representations. A connected component is said to have trivial valuations when every arrow valuation is trivial.

The authors' conjecture. The connected components of \text{\mathit\Gamma}_{{\rm rep}(Q)} containing representations not in $\overline{\mystrut(5,0) {\rm rep\hspace{-1pt}}}\,(Q)$ are connected subquivers of

$$\cdots \to \circ \to \circ \to \circ \to \cdots$$

and all have trivial valuations.

The preceding results show that components containing only representations in the finitely presented subcategory are components of the Auslander–Reiten quiver of ${\rm rep}(Q)$. The conjecture describes the remaining components, which arise from representations outside that subcategory, and asserts that they are infinite linear components with no nontrivial valuations.