Conjecture F_const for constant Brauer classes

Let $k$ be a number field and let $\pi_+=(M,(L_m)_{m\in M},(b_m)_{m\in M},(K_m)_{m\in M})\in{\mathscr P}_+$. Let $B_{\mathrm{const}}\subseteq B$ be the subgroup generated by the corestricted classes associated with characters in $C_{m,\mathrm{const}}=C_m\cap\operatorname{Im}(H^1(k(m),{\mathbf Q}/{\mathbf Z})\to H^1(L_m,{\mathbf Q}/{\mathbf Z}))$. For $c\in U(k)$, let $W_c({\mathbf A}_k)^{B_{\mathrm{const}}}$ denote the adelic points orthogonal to the image of $B_{\mathrm{const}}$ on the fibre.

Conjecture F_const. The subset

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

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

Conjecture F_const is an intermediate formulation used to compare Conjectures F and F+. The source notes that, under suitable abelianness assumptions, it is implied by Schinzel's hypothesis $(\mathrm{HH}_1)$; its general status is not specified.