The E4 Weyl group symmetry and invariant tiling conjecture

Let $g=(A_1A_2)^2$ with $s$ irrational, and let the associated double lattice PET from the compactification theorem be defined on a compactified space with invariant foliation. Denote its almost everywhere defined invariant polytope tiling by $\widehat T_s$, and let $L_z$ be the leaf of the invariant foliation through $z\in\mathbb{C}$. E4 Weyl group tiling conjecture. The double lattice PET has an almost everywhere defined invariant tiling $\widehat T_s$ by polytopes; each polytope has order $192$ symmetry induced by the action of the $E4$ Weyl group; and, for every $z\in\mathbb{C}$, $L_z\cap\widehat T_s$ is a refinement of the tiling of $\mathbb{C}$ by periodic islands of $g_s$. This conjecture describes the expected symmetry and leafwise relation between the compactified tiling and the original periodic-island tiling. The supplied text gives no evidence that it has been proved or disproved.