The geometric model conjecture for dimer tree algebras

Let $B$ be a dimer tree algebra, let $\mathcal{S}$ be its checkerboard polygon with $2N$ vertices, and let $\gamma$ be a 2-diagonal. For the projective modules $P_0(\gamma)$ and $P_1(\gamma)$ determined by the degree-zero and degree-one radical-line crossings of $\gamma$, let $\Omega$ denote the syzygy functor. Geometric model conjecture. For every 2-diagonal $\gamma$ there exists a morphism

$$f_\gamma\colon P_1(\gamma)\to P_0(\gamma)$$

such that $\gamma\mapsto\operatorname{coker}f_\gamma$ induces an equivalence

$$F\colon\operatorname{Diag}(\mathcal{S})\to\underline{\operatorname{CMP}}\,B.$$

Under this equivalence, $\rho(i)$ corresponds to $\operatorname{rad}P(i)$, clockwise rotation $R$ corresponds to $\Omega$, $R^2$ corresponds to $\tau^{-1}=\Omega^2$, 2-pivots correspond to irreducible morphisms, and meshes correspond to Auslander–Reiten triangles. The conjecture is the paper's central categorical model for syzygies; no resolution evidence is supplied in the input.