Torus conjecture for toric homotopy path algebras

Let $\mathcal{X}$ be a proper Deligne–Mumford toric stack, let $A_{\mathcal{X}}$ be a toric FSEC homotopy path algebra associated to it, and let $X_{A_{\mathcal{X}}}$ be the associated cellular complex. Let $\mathbb{T}$ denote a real torus.

Torus conjecture. The complex $X_{A_{\mathcal{X}}}$ is homotopic to a real torus $\mathbb{T}$.

Examples in the paper show this behavior for weighted projective stacks such as $\mathbb{P}(1:1:n)$, and the preceding proposition establishes that the associated complex has Euler characteristic zero. The general assertion for proper Deligne–Mumford toric stacks remains open in the supplied text.