Tiling by the ideal right-angled octahedron

Prove that every ideal, right-angled polyhedron assembled from copies of the Scharlau–Walhorn polyhedron SW#1{2} is tiled by the ideal, right-angled hyperbolic octahedron.

Background

The paper classifies ideal, right-angled polyhedra assembled from the 49 Scharlau–Walhorn arithmetic Coxeter polyhedra. For SW#1{2}, Proposition \ref{Bipyramid} proves that every such polyhedron is tiled by smaller pieces derived from the octahedron: the triangular bipyramid \frac14\cO, the intermediate polyhedron \frac12\cO, and \cO itself. The unresolved strengthening is that the decomposition should always be obtainable directly from copies of the full ideal, right-angled octahedron \cO.

Establishing this would sharpen the classification by showing that the octahedron, square antiprism, and rhombicuboctahedron are the unique minimal ideal, right-angled hyperbolic polyhedra with arithmetic reflection groups.