Canonical Jordan recoverability criterion for subcategories of representations of $A_n$-type quivers

Let $d4ac$ be an $A_n$-type quiver, and let $d524$ be a subcategory of $d53d\operatorname{rep}(d4ac)$. For strings, write $\mathsf{Supp}_0(\rho)$ for the relevant vertex support and $\mathsf{Supp}_1(\rho)$ for the relevant arrow support. A pair of strings is associated to indecomposable representations of $d524$ when the corresponding indecomposable representations belong to $d524$. Canonical Jordan recoverability conjecture. The subcategory $d524$ is canonically Jordan recoverable if and only if, for every pair of strings $(\rho,\nu)$ associated to indecomposable representations of $d524$, there is no arrow $\alpha\in Q_1$ such that

$$s(\alpha)\in\mathsf{Supp}_0(\rho),\qquad t(\alpha)\in\mathsf{Supp}_1(\nu),\qquad \alpha\notin\mathsf{Supp}_1(\rho)\cup\mathsf{Supp}_1(\nu).$$

This proposed characterization is intended to describe all canonically Jordan recoverable subcategories of module categories over gentle algebras in the $A_n$-type case. The source presents it as a conjecture motivated by further work on $A_n$-type quivers; its resolution is not supplied.