  We show that the simple elements of the dual Garside structure of an Artin
group of type $D_n$ are Mikado braids, giving a positive answer to a conjecture
of Digne and the second author. To this end, we use an embedding of the Artin
group of type $D_n$ in a suitable quotient of an Artin group of type $B_n$
noticed by Allcock, of which we give a simple algebraic proof here. This allows
one to give a characterization of the Mikado braids of type $D_n$ in terms of
those of type $B_n$ and also to describe them topologically. Using this
topological representation and Athanasiadis and Reiner's model for noncrossing
partitions of type $D_n$ which can be used to represent the simple elements, we
deduce the above mentioned conjecture.
