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?

Edit: As pointed out by @Donu Arapura in the comments, the statement is trivially wrong if we do not assume that $k$ is algebraically closed. So we should assume $k$ is algebraically closed.