Here, a reflection quotient is a surjective homomorphism between Coxeter groups mapping reflections to reflections. A reflection quotient $C(\Delta) \to C(\Gamma)$ is proper if it is not injective and $\Gamma$ is not the single vertex graph.
An example of a proper reflection quotient is the homomorphism $C(D_n) \to C(A_{n-1})$ identifying two leaves of $D_n$. Are there any other reflection quotients $C(D_n) \to C(\Gamma)$? If yes, which ones?
Note that $C(A_{k-1}) \cong S_k$ does not have any proper reflection quotients for any $k$. This can be seen by noting that if $k \neq 4$ the only proper normal subgroup of $S_k$ is simple of index two, and if $k = 4$ the only other quotient is in fact $S_3$, but not in a reflection-preserving manner. Because $C(A_{n-1})$ is not only a reflection quotient but also a quite big reflection subgroup (i.e. a subgroup generated by reflections) of $C(D_n)$, it seems likely that the answer to my question is no.
Literature about the topic of what I called reflection quotients would be greatly appreciated, as I did not find any.
The answer is indeed no, unless $n = 4$. This is shown in the article The normal subgroups of finite and affine Coxeter groups by George Maxwell. In Table 3 he lists the reflection quotients of $D_n$ (which are just the mentioned ones), and an additional reflection quotient $C(A_2)$ in the case $n = 4$.
See also my answer on MO for a more detailed and more general answer. That thread also contains answers for other classes of Coxeter groups.