Torus and rationality conjecture for homotopy path algebras

Let $\mathcal{X}$ be a proper Deligne–Mumford stack of dimension $n$ with a full strong exceptional collection whose endomorphism algebra $A$ is a homotopy path algebra. Let $X_A$ be the associated cellular complex, and let $\mathbb{T}^n$ be the real $n$-torus.

Torus and rationality conjecture. There exists a subcomplex $Y \subseteq X_A$ homotopic to $\mathbb{T}^n$. Furthermore, $\mathcal{X}$ is rational, with local systems on $Y$ corresponding to points on $\mathcal{X}$, and $X_A$ is homotopic to a torus if and only if $\mathcal{X}$ is toric.

This conjecture proposes a topological model for proper Deligne–Mumford stacks admitting such exceptional collections and relates torus topology of $X_A$ to toricness. No resolution or partial result for the full assertion is given in the supplied text.