On the Bonnaf\'e--Dat--Rouquier Morita equivalence

We prove that the cohomology group of a Deligne-Lusztig variety defines a Morita equivalence in a case which is not covered by the argument by Bonnaf'e, Dat and Rouquier, specifically we consider the situation for semisimple elements in type $D$ whose centralizer has non-cyclic component group.