The non-all-net conjecture for the 24-cell, 120-cell, and 600-cell

The 24-cell, the 120-cell, and the 600-cell are regular four-dimensional polytopes.

All-net conjecture. The 24-cell, the 120-cell, and the 600-cell fail to be all-net.

The all-net property asks whether every net, or unfolding, of a polytope has non-overlapping facets. The corresponding property is known for the Platonic solids, the simplex, and the cube, while the orthoplex fails it except in dimension four. The claim concerns the three remaining regular polytopes whose all-net property had not been studied; its resolution is not given here.