Gritsenko–Nikulin's completeness conjecture for rank-3 hyperbolic generalized Cartan matrices

Let $A$ be a hyperbolic generalized Cartan matrix of rank $3$ and elliptic type, with a lattice Weyl vector, whose geometric realization is twisted to a symmetric generalized Cartan matrix. Write $\lambda_i$ for its twisting coefficients and $G(A)$ for its geometric realization. Gritsenko and Nikulin's theorem lists all such matrices satisfying $\lambda_i\leq 12$. Gritsenko–Nikulin's completeness conjecture. Table 1 gives the complete list of such matrices $G(A)$; equivalently, the inequality $\lambda_i\leq 12$ can be omitted from the theorem. Gritsenko and Nikulin support this with the finiteness of the collection in rank at least $3$ and computations showing no new solutions for twisting coefficients between $6$ and $12$. The paper proposes a four-step approach to the conjecture and gives a complete solution for the first step.