This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is:

Is the Brauer group $\text{Br}(X)$ of the

rationalsurface $X$ always trivial?

This might be true and well-known in a more general setting but I can't find a good reference. Let me give some context and show how the question arose. There is an exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow \text{Cl}(X) \rightarrow \bigoplus_{x \in X^{\text{sing}}} \text{Cl}(\mathcal{O}_{X,x}) \rightarrow H^{2}(X,\mathcal{O}_{X}^{\ast}) \rightarrow 0$$ of abelian groups. If $X$ has at most ADE singularities one obtains $$\text{Br}(X) = H^{2}(X,\mathcal{O}_{X}^{\ast})_{\text{tor}} = H^{2}(X,\mathcal{O}_{X}^{\ast})$$ where the first equality follows from $X$ being a normal surface (see here) and the second one from $\bigoplus_{x \in X^{\text{sing}}} \text{Cl}(\mathcal{O}_{X,x})$ being a torsion group (because $X$ is ADE).

If it was true that the Brauer group is trivial in the rational case I could make more precise statements about $\text{Cl}(X)$ and $\text{Pic}(X)$.

Is there a reference for this? I'm new to this and I'm not even entirely sure if the above is right. So please correct me if I'm wrong. Thanks in advance.

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