Special vertices in the Grünbaum–Shephard–Geis catalogue

Determine which irreducible simplicial arrangements in the Grünbaum–Shephard–Geis catalogue of arrangements in real projective three-space admit a special vertex.

Background

The paper establishes that the irreducible crystallographic Coxeter arrangements of types A4 and B4 admit a special vertex, while those of types D4 and F4 do not. It also exhibits a deletion chain of irreducible simplicial arrangements inside B4 with this property.

The authors then ask for a classification across the broader Grünbaum–Shephard–Geis catalogue. Resolving this question would determine how widespread the special-vertex property is among known irreducible simplicial arrangements in projective three-space.