Grünbaum–Cuntz conjecture for rank-3 simplicial arrangements

A simplicial arrangement is a hyperplane arrangement whose every region is a simplicial cone. Two arrangements are isomorphic when their associated fans are combinatorially isomorphic. The Grünbaum–Cuntz catalog is the catalog of the known rank-$3$ simplicial arrangements.

Grünbaum–Cuntz conjecture. Every simplicial arrangement of rank $3$ is isomorphic to an arrangement in the Grünbaum–Cuntz catalog.

This is the completeness conjecture underlying the paper's conditional finiteness and projective-uniqueness conclusions. The paper proves computational results for the catalog but does not establish that the catalog contains every rank-$3$ simplicial arrangement.