Loughran's conjecture for counting points orthogonal to a Brauer subgroup

Let $X$ be a smooth projective variety over a number field $F$ satisfying the stated assumptions, and let $H$ be an anticanonical height function. Let $\mathscr{B}\subset\operatorname{Br}F(X)$ be finite, and suppose there is an $x\in X(F)$ at which every $b\in\mathscr{B}$ is defined and satisfies $b(x)=0$. For a divisor $D\in X^{(1)}$, let $\operatorname{\partial}_D(\mathscr{B})$ denote the residue subgroup, and set

$$\Delta_X(\mathscr{B})=\sum_{D\in X^{(1)}}\left(1-\frac{1}{|\operatorname{\partial}_D(\mathscr{B})|}\right).$$

Loughran's conjecture. There exist an open subset $U\subset X$ with $\mathscr{B}\subset\operatorname{Br}U$ and a constant $c_{U,\mathscr{B},H}>0$ such that

$$N(U,H,\mathscr{B},B)\sim c_{U,\mathscr{B},H}B\frac{(\log B)^{\rho(X)-1}}{(\log B)^{\Delta_X(\mathscr{B})}},\qquad B\to\infty.$$

The conjecture predicts how Brauer-group conditions alter the usual Peyre-type point-counting asymptotic. Its resolution status is not specified in the supplied text.