Let $(W,S)$ be a Coxeter group, $Isubseteq S$ a subset of simple reflections, and $W_I subseteq W$ the corresponding parabolic subgroup (we could also assume $|W_I|<infty$, if needed).

Let also $t_1,t_2in W$ be two reflections (i.e. elements in $W$ conjugated to some $s_1,s_2 in S$ respectively) such that $t_1t_2in W_I$ but $t_inotin W_I$, $i=1,2$.

Is it then true that the only possibility is $t_1t_2=e$ (i.e. $t_1=t_2$)?

If I look at the geometric picture with alcoves and reflection hyperplanes this seems to me to be true, but I don’t know how to prove it in full generality via geometric arguments. I rather tried with some combinatoric method (e.g. using the strong exchange condition or some result about parabolic double cosets) but I have succeeded only in the case $t_i in S$ for at least one $i$.

Thanks a lot for any comment about that!