The projective-index classification conjecture for finite linear group actions on $\mathbb{C}^2$

Let $G$ be a finite group with a linear action on $\mathbb{C}^2$. Its projective indices $\tau_P^G$ and associated canonical group are among the following possibilities: $\{(0,0)\}$ for a cyclic group; $\{(0,0),(1,1),(3,3),\ldots,(m^*,m^*)\}$ for a binary dihedral group of order $4m$; $\{(0,0),(1,1),(2,2),(5,5)\}$ for a binary tetrahedral group; $\{(0,0),(1,1),(2,2),(3,3),(5,5),(7,7),(11,11)\}$ for a binary octahedral group; or $\{(p,p):p=0,1,2,3,4,5,7,8,9,11,13,14,17,19,23,29\}$ for a binary icosahedral group.

Projective-index classification conjecture. Every finite group with a linear action on $\mathbb{C}^2$ has one of these five projective-index sets and associated canonical groups.

The paper explains that the list of canonical groups is known, and that the remaining issue is excluding further projective indices $(p,p)$ with $p>100$, beyond the range computed. Thus the classification is supported by computation but is not established in full.