The quasi-isomorphism conjecture for arithmetic graphs and the plaid model

Let $p/q$ be the rational parameter, let $\Gamma(p/q)$ be its arithmetic landscape, and let $T_{p/q}$ be the affine map carrying the plaid model's fundamental domain to the arithmetic graph's fundamental domain. Consider the polygons of $T_{p/q}^{-1}(\Gamma_{p/q})$ and the polygons of the plaid model in their respective fundamental domains. Quasi-isomorphism conjecture. There is a bijection between these two polygon sets such that each polygon in one model lies within a $2$-tubular neighborhood of its corresponding polygon in the other. This conjecture asserts that the arithmetic graph and plaid model have the same coarse polygonal geometry, up to a uniformly bounded discrepancy; the surrounding discussion presents it as the speculative correspondence underlying the relationship between special outer-billiards orbits and the plaid model.