The stabilizer conjecture for the sets $W_l'$ in three-reflection presentations

Let $D_n$ be a dihedral group with cyclic subgroup $C_n$, and let $W_l'$ be the subset arising at level $l$ in the three-reflection presentation construction used in the paper. Denote its stabilizer by

$$\operatorname{Stab}(W_l')=\{c\in C_n:W_l'+c=W_l'\}.$$

Stabilizer conjecture. The stabilizer of $W_l'$ is always either

$$\{1\}\quad\text{or}\quad C_n.$$

The conjecture is suggested by the proof of the preceding theorem, but the source gives no resolution or further context.