LRS asymptotic formula for rational points on multiple conics

Let $X$ be a smooth, projective, geometrically integral, weak Fano variety over $\mathbb{Q}$. Let $U \subseteq X$ be an open subset and let $\mathscr{B} \subseteq \operatorname{Br} U$ be a finite subset. Assume that $\operatorname{Br} X = \operatorname{Br}_1 X$, that $U(\mathbb{Q})_{\mathscr{B}} \neq \varnothing$, and that either $\operatorname{rk}(\operatorname{Pic} X) = 1$ or $\mathbb{Q}[U]^\times = \mathbb{Q}^\times$. LRS conjecture. There exists a thin subset $\Omega \subset X(\mathbb{Q})$ such that

$$\#\{ x \in U(\mathbb{Q})_{\mathscr{B}}: H(x) \leq B, x \notin \Omega \} \sim c_{\textup{LRS}} B(\log B)^{\operatorname{rk}(\operatorname{Pic} X) - \Delta(\mathscr{B}) - 1},$$

where

$$\Delta(\mathscr{B}) = \sum_{D \in X^{(1)}} \left(1 - \lvert\partial_D(\langle\mathscr{B}\rangle)\rvert^{-1}\right),$$ $$c_{\textup{LRS}} = \frac{\theta(X) \cdot \lvert\operatorname{Br}_{\mathrm{Sub}}(X,\mathscr{B})/\operatorname{Br} \mathbb{Q}\rvert \cdot \tau_{\mathscr{B}}\left(X(\mathbb{A}_{\mathbb{Q}})_{\mathscr{B}}^{\operatorname{Br}_{\mathrm{Sub}}(X,\mathscr{B})}\right)}{\Gamma(X,\mathscr{B})} \cdot \prod_{D \in X^{(1)}} \eta(D)^{1-\lvert\partial_D(\langle\mathscr{B}\rangle)\rvert^{-1}},$$

and

$$\Gamma(X,\mathscr{B}) = \begin{cases} \displaystyle \prod_{D \in X^{(1)}}\Gamma(\lvert\partial_D(\langle\mathscr{B}\rangle)\rvert^{-1}) & \text{if } \operatorname{rk}(\operatorname{Pic} X) = 1, \\ \displaystyle \Gamma\left(\operatorname{rk}(\operatorname{Pic} X) - \Delta(\mathscr{B})\right) & \text{if } \mathbb{Q}[U]^\times = \mathbb{Q}^\times. \end{cases}$$

This is an expected Manin-type asymptotic for rational points satisfying the Brauer conditions on weak Fano varieties. The supplied text does not establish the assertion or provide evidence resolving its status.