Várilly-Alvarado's strong uniform boundedness conjecture for K3 Brauer groups

Fix a positive integer $n$ and a primitive lattice embedding

$$\Lambda\hookrightarrow\Lambda_{K3}:=U^{\oplus 3}\oplus E_8^{\oplus 2}.$$

Let $X$ be a K3 surface over a number field of degree $n$ such that the geometric Néron–Severi lattice $\operatorname{NS}\overline{X}$ is isomorphic to $\Lambda$ as an abstract lattice. Strong uniform boundedness conjecture. There is a constant $C(n,\Lambda)$, independent of $X$, such that

$$\#\bigl(\operatorname{Br}X/\operatorname{Br}_0X\bigr)\leq C(n,\Lambda).$$

The conjecture asserts uniformity after fixing both the degree of the field of definition and the geometric Néron–Severi lattice; the paper studies explicit bounds in important cases, but the stated general uniform assertion remains open.