For an abelian variety $A$ over a global field $K$, $\mathcal{A}$ its Néron model on $C$, either a smooth projective geometrically irreducible curve over a finite field, or the spectrum of the ring of integers of a number field, is it true that:

$$\text{H}^2(A_{ét},\mathbf{G}_m)/\text{H}^2(K,\mathbf{G}_m) = \text{H}^2(\mathcal{A}_{et},\mathbf{G}_m)/\text{H}^2(C_{et},\mathbf{G}_m)\ ?$$

In other words, is the following identity between cohomological Brauer groups true?

$$\text{Br}(A)/\text{Br}(K) = \text{Br}(\mathcal{A})/\text{Br}(C).$$