Várilly-Alvarado's uniform Brauer-group bound for K3 surfaces

Let $L$ be an even hyperbolic lattice and let $X/K$ be a K3 surface over a number field such that

$$\operatorname{NS}(\overline{X}) \cong L.$$

Várilly-Alvarado's conjecture. The order

$$|\operatorname{Br}(\overline{X})^{G_K}|$$

can be bounded only in terms of $[K:\mathbb{Q}]$.

Such a uniform bound would make the characterization of the Noether–Lefschetz locus effective. The statement is presented as a conjecture; no resolution is supplied here.