The cyclicity conjecture for finite groups from skew-plane configurations

Let $\mathcal L=\{L_\infty,L_0,L_1,\dots,L_r\}\subset \mathbb{P}^{2n+1}_{\mathbb C}$, with $n>1$, and let $G_{\mathcal L}$ be the associated group generated by the transformations determined by the configuration.

Cyclicity conjecture. If $G_{\mathcal L}$ is finite, then $G_{\mathcal L}$ is cyclic.

Examples in characteristic zero show that several natural non-cyclic candidates, including examples with finite-order generators, nevertheless generate infinite groups. These observations motivate the conjecture that the cyclic case exhausts all finite possibilities.