Pure matching curves and finite discrete indecomposable modules

Assume that the ground ring $k$ is a field. Let $\mathscr{J}_\mathcal{T}$ and $\mathscr{G}_\mathcal{T}$ be the algebras appearing in the construction, and let

$$\nu:\mathcal{D}(\mathscr{J}_\mathcal{T})^\heartsuit\simeq \mathcal{D}(\mathscr{G}_\mathcal{T})^\heartsuit \hookrightarrow \mathcal{D}(\mathscr{G}_\mathcal{T})$$

be the inclusion of the heart. A finite pure matching curve is a pure matching curve with finitely many segments, and $M_\gamma$ denotes the associated $\mathscr{G}_\mathcal{T}$-module; a finite discrete indecomposable $\mathscr{J}_\mathcal{T}$-module is an indecomposable module of the stated discrete type. Pure matching-curve classification conjecture. The functor $\nu$ restricts to a bijection between the isomorphism classes of $\mathscr{G}_\mathcal{T}$-modules of the form $M_\gamma$, with $\gamma$ a finite pure matching curve, and the isomorphism classes of finite discrete indecomposable $\mathscr{J}_\mathcal{T}$-modules. This conjectural bijection would identify the indecomposable objects of the heart on the geometric side with finite discrete indecomposable modules; the source gives no resolution, so the conjecture remains open.