Strong uniform boundedness conjecture for Brauer groups of K3 surfaces

Let $k$) be a number field with $[k:\mathbb{Q}]\leq d$, let $L\hookrightarrow\Lambda$ be a primitive sublattice, and let $X$ be a K3 surface over $k$ such that $\operatorname{Pic}(\overline{X})\simeq L$. The quotient $\operatorname{Br}(X)/\operatorname{Br}_0(X)$ is finite, where $\operatorname{Br}_0(X)$ is the subgroup of constant algebras.

Strong uniform boundedness conjecture. There is a constant $B=B(d,L)$, independent of $X$, such that

$$\#\frac{\operatorname{Br}(X)}{\operatorname{Br}_0(X)}<B.$$

This conjecture predicts uniform boundedness in fixed-degree number fields and fixed geometric Picard lattice. The algebraic quotient $\operatorname{Br}_1(X)/\operatorname{Br}_0(X)$ is already known to be uniformly bounded, so the substantive issue is the transcendental Brauer group. The source gives no resolution status.