Derived Reid's recipe conjecture for consistent dimer models

Let $Q_0$ be the vertex set of the dual quiver, let $\theta$ be a generic stability parameter, and let $\theta$ denote the corresponding moduli space with fan $\Sigma_\theta$. For a nonzero vertex $i\in Q_0$, let $\Psi_\theta(S_i)$ be the associated derived object and write $\mathcal{F}:=H^{-1}(\Psi_\theta(S_i))$. Interior line segments $\tau\in\Sigma_\theta(2)$ and lattice points $\rho\in\Sigma_\theta(1)$ are marked by vertices according to the combinatorial Reid's recipe. Derived Reid's recipe conjecture. For every nonzero vertex $i\in Q_0$, $\mathcal{F}\neq 0$ if and only if $i$ marks two or more interior line segments in $\Sigma_\theta$. In that case, the support of $\Psi_\theta(S_i)$ is the union of all torus-invariant divisors $D_\rho$ such that two line segments $\tau$ in $\Sigma_\theta$ containing $\rho$ are marked by $i$. This would complete the description of the supports of the derived objects associated to nonzero vertices, extending the cases where a vertex marks lattice points or a unique line segment; the conjecture is presented as open in the source.