Uncountably many maximal subgroups in hyperbolic triangle groups

Let $\Gamma$ be a hyperbolic triangle group. A subgroup has infinite index if its index in $\Gamma$ is infinite, and two subgroups are conjugate when one is obtained from the other by conjugation in $\Gamma$. Uncountability conjecture. Every hyperbolic triangle group has uncountably many conjugacy classes of maximal subgroups of infinite index. The paper establishes this for the groups $\Delta(2,3,r)$ and $\Delta[2,3,r]$ when $r\geq 7$, but the assertion for every hyperbolic triangle group remains open.