Conjecture F+ for fibrations with controlled Brauer classes

Let $k$ be a number field. Let $\pi_+=(M,(L_m)_{m\in M},(b_m)_{m\in M},(K_m)_{m\in M})\in{\mathscr P}_+$, determining a variety $W$, a smooth morphism $p:W\to{\mathbf P}^1_k$, and $U={\mathbf P}^1_k\setminus M$. Let $B\subseteq\operatorname{Br}(p^{-1}(U))$ be the finite subgroup generated by the classes associated with the characters in $C_m=\operatorname{Ker}(H^1(L_m,{\mathbf Q}/{\mathbf Z})\to H^1(K_m,{\mathbf Q}/{\mathbf Z}))$. For $c\in U(k)$, let $W_c({\mathbf A}_k)^B$ be the adelic points orthogonal to the image of $B$ under restriction to $\operatorname{Br}(W_c)$.

Conjecture F+. The subset

$$\bigcup_{c\in U(k)}W_c({\mathbf A}_k)^B$$

is dense in $W({\mathbf A}_k)$.

This is proposed as an improved substitute for Conjecture F and incorporates a prescribed Brauer–Manin condition on the fibres. Its status is not specified in the source.