The positive Euler characteristic conjecture for two-dimensional Coxeter groups

Let $G$ be a finitely generated infinite Coxeter group of dimension at most $2$. Its Euler characteristic is denoted by $4\chi(G)4$. Positive Euler characteristic conjecture. If

$$\chi(G)>0$$

then $G$ is incoherent. This would generalize the paper's incoherence results for Coxeter groups with positive Euler characteristic, but the statement is presented as a conjectural consequence rather than proved here.