Determine the prototile count for the 4D Elser–Sloane quasicrystal (ESQC)

Determine the number of distinct prototiles required to tile the four-dimensional Elser–Sloane quasicrystal derived from the E8 lattice, and, if possible, characterize these prototiles explicitly.

Background

In discussing lower-dimensional quasicrystals (e.g., the Fibonacci chain and Penrose tiling), the authors note that the number of tiles is well understood, often involving just two tile types. However, for the four-dimensional ESQC derived from E8, they point out that the number of tiles is not well known, even though the 600-cell is identified as a building block arising at multiple scales.

This indicates a concrete unresolved structural question about the ESQC’s prototile set that is relevant for their cosmological modeling of quasicrystalline growth and rescaling.