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 $L$?