Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$. 

For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed point $x\in X$, my question is: 
>>Could we always find a Zariski open neighborhood $U$ of $x$ (which depends on $\alpha$), such that $\alpha|_U$ is trivial in $H^2_{et}(U,\mathcal{O}_U^*)$? 

If not, 

 1. Is there any counter-example? 
 2. Is the same statement true if we restrict ourselves to the Brauer group rather than the cohomological Brauer group, i.e. elements in $H^2_{et}(X,\mathcal{O}_X^*)$ that comes from an Azumaya algebra?