One fundamental conjecture on the Brauer group is that $\operatorname{Br}(X)$ is *finite* for $X/\operatorname{Spec} \mathbf{Z}$ proper. By class field theory (the theorem of Albert–Brauer–Hasse–Noether), this is true for $\dim{X} = 1$. For varieties $X$ over finite fields, the finiteness of $\operatorname{Br}(X)[\ell^\infty]$ is equivalent to the surjectivity of the cycle class map $\operatorname{Pic}(X) \otimes \mathbf{Z}_\ell \to \operatorname{H}^2(X,\mathrm{Z}_\ell(1))$ (the Tate conjecture in dimension $1$). The Tate conjecture in dimension $1$ over finite fields is known for smooth projective curves (trivial), Abelian varieties (Tate) and K3 surfaces (at least for characteristic $> 3$), and hence for products of such varieties.

Are there results on the finiteness (of an $\ell$-primary part) of the Brauer group of schemes

flatand proper over $\operatorname{Spec} \mathbf{Z}$ of dimension $> 1$?

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