# Artin groups of type $D_n$ as mapping class groups?

According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $$A(D_n$$) of type $$D_n$$ may be realized as an index 2 subgroup of the orbifold fundamental group of $$\{x \in L^n | \forall i \neq j, x_i \neq x_j\}$$, where $$L$$ is the orbifold consisting of a disc with one cone point of order $$2$$.

My question is the following: is there a natural representation of the Artin group $$A(D_n)$$ as some mapping class group of $$L$$, similar to the Birman theorem realizing the classical braid group as a mapping class group of a punctured disc? Or is there a natural action of the Artin group $$A(D_n)$$ on some complex of curves/arcs on the orbifold $$L$$?