# 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 $$L$$?

Perron-Vannier prove that the Artin group $$A(D_n)$$ "geometrically embeds" into the mapping class group of a surface, i.e., that there is a surface $$\Sigma$$ with boundary (and no punctures) and a faithful (albeit not surjective) representation $$A(D_n) \to \rm{Mod}(\Sigma)$$ mapping the standard generators to Dehn twists.
The surface $$\Sigma$$ can be chosen as follows. Take a disc $$\Delta$$ and embedded arcs in $$\Delta$$ that intersect in the pattern of $$D_n$$ (edges in $$D_n$$ correspond to intersection points and vertices to arcs). Close up each arc with a band to form $$\Sigma$$. The Dehn twists along the closed arcs then generate a subgroup of $$\rm{Mod}(\Sigma)$$ isomorphic to $$D_n$$. In fact, the mentioned Dehn twists map to the standard generators of $$A(D_n)$$.
Labruère later studied the same construction of $$\Sigma$$, but taking the intersection pattern of a cycle $$\widetilde A_{n-1}$$ rather than $$D_n$$. The kernel of the resulting (still not surjective) representation $$A(\widetilde A_{n-1}) \to \rm{Mod}(\Sigma)$$ is generated by the so-called "cycle relation", and one can show that the quotient of $$A(\widetilde A_{n-1})$$ by that relation is again isomorphic to $$A(D_n)$$ (for example, using Baader-Lönne's results).