Strong uniform boundedness conjecture for Brauer groups of K3 surfaces without fixed Picard lattice

Let $k$ be a number field with $[k:\mathbb{Q}]\leq d$, and let $X/k$ be a K3 surface. 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. If $X$ is a K3 surface over a number field of bounded degree $d$, then

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

is bounded in terms of $d$, independently of $X$.

This is a stronger version of the lattice-polarized boundedness conjecture, dispensing with the fixed lattice $L$. The source presents it as a conjecture inspired by a conjecture of Shafarevich and gives no resolution status.