Let $(W,S)$ be an irreducible affine Coxeter system of rank $n \geq 3$ (affine for instance as in the sense of Chapter 4 of Humphreys "Reflection groups and Coxeter groups").

Let $t_1,\ldots, t_n$ be a set of (not necessarily simple) reflections such that $W= \langle t_1, \ldots , t_n \rangle$. Further assume that $\langle t_1, \ldots , t_{n-2} \rangle$ is finite.

Does there exist a reflection $r \in \langle t_{n-1}, t_n \rangle $ such that $\langle t_1, \ldots, t_{n-2},r \rangle$ is finite as well?

An easy example: $(W, \{s_1,s_2,s_3\})$ affine of type $\widetilde{A}_2$. Then obviously we have $W=\langle s_1, s_2s_3s_2,s_2 \rangle$ and $\langle s_1 \rangle$ is finite. While $\langle s_1, s_2s_3s_2 \rangle$ is infinite, we have that $\langle s_1,s_2 \rangle$ is finite.

It is well known that a Coxeter group is infinite iff it contains an infinite dihedral group. So equivalently, the question is whether there exists a reflection $r \in \langle t_{n-1}, t_n \rangle $ such that $\langle t_1, \ldots, t_{n-2},r \rangle$ does not contain an infinite dihedral subgroup?