The braid group Bn is not a quotient of a quasi-Coxeter interval group of type Dn

We prove that, for $n\ge 5$, an interval group associated with a proper quasi-Coxeter element of the Coxeter group of type $D_n$ admits no surjective homomorphism onto the braid group on $n$ strands. In particular, this provides an alternative proof that such a group is not isomorphic to the Artin group of type $D_n$. The proof relies on techniques from the theory of mapping class groups, and a large part of the paper provides an exposition of this theory for non-specialists.