For any scheme $X$, let $\operatorname{Br}X$ denote the (Azumaya) Brauer group of $X$, namely the Morita equivalence classes of Azumaya $\mathcal{O}_{X}$-algebras.

Is the functor $$\operatorname{Br} : \operatorname{Sch}^{\operatorname{op}} \to \operatorname{Ab}$$ sending $X \mapsto \operatorname{Br}X$ a sheaf for the Zariski topology on $\operatorname{Sch}$?

In other words, if $X = \bigcup_{i \in I} U_{i}$ is a Zariski open covering of a scheme $X$, is the sequence $$0 \to \operatorname{Br}X \to \prod_{i \in I} \operatorname{Br}U_{i} \to \prod_{i_{1},i_{2} \in I} \operatorname{Br}(U_{i_{1}} \cap U_{i_{2}}) $$ exact?

Thoughts: My naive guess is "no" since $\operatorname{Br}$ is not an etale sheaf and $\operatorname{Pic}$ is not a Zariski sheaf. Exercise 8(f) of http://www-personal.umich.edu/~bhattb/teaching/mat731fall2011/ex6.pdf shows that the functor $U \mapsto \mathrm{H}_{et}^{2}(U,\mathbb{G}_{m})$ is a Zariski sheaf if $X$ is regular Noetherian. See also the discussion on Milne's "Etale Cohomology", IV, Remark 2.10.