Let X$X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{\'et}}(X, \mathbb{G}_m)$$\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (sometimes people define the Brauer group as the subgroup of torsion elements). A fundamental fact in the theory of Brauer groups is that for every irreducible divisor $D \subset X$ with smooth locus $D^{\text{sm}}$ and complement $U := X \setminus D$ there exists a residue map $\partial_D: \text{Br}(U) \to H^1(D^{\text{sm}}, \mathbb{Q}/\mathbb{Z})$ such that the sequence $0 \to \operatorname{Br}(X) \to \operatorname{Br}(U) \xrightarrow{\partial_D} H^1(D^{\text{sm}},\mathbb{Q}/\mathbb{Z})$ is exact. This is known as Grothendieck purity, see e.g. Theorem 3.7.2 of the following book: The Brauer-Grothendieck Group.
For a paper I'm writing I need an analogue of the above statement for algebraic stacks (Deligne-Mumford stacks are also fine but algebraic stacks would be better). The main technical ingredient in the proof of Grothendieck purity is cohomological purity for finite coefficients. This fact generalizes to stacks, see Proposition 4.9.1 of Laszlo and Olsson's The six operations for sheaves on Artin stacks I: Finite Coefficients. It thus seems like one should be able to use the proof for varieties also for stacks.
This proof would be quite involved so if possible I would prefer to just cite something. Does anybody know if Grothendieck purity has been proven in the literature? The case when $X$ is Deligne-Mumford and has dimension 2 is Proposition 2 of Stable rationality and conic bundles but I need a version in arbitrary dimensions. Note that by this mathoverflow answer one may assume that D is smooth.
