The $E''$-family $H$-moment conjecture

Let $Q$ be a global field, let $\Gamma$ be a finite group, and let $T$ be a finite set of places with prescribed decomposition subgroups $(Gamma_v)_{v otin T}$. Let $E”_{Gamma,(Gamma_v)_{v otin T}}(D,Q)$ consist of the corresponding $Gamma$-extensions satisfying the cyclotomic-disjointness condition but not necessarily the roots-of-unity condition used to define $E'$. Let $H$ be a finite admissible $Gamma$-group.

$E”$-family $H$-moment conjecture. The filtered $H$-moment conjecture remains true after every occurrence of $E'_{Gamma,(Gamma_v)_{v otin T}}(D,Q)$ is replaced by $E”_{Gamma,(Gamma_v)_{v otin T}}(D,Q)$.

This is proposed as a way to retain extensions excluded by the roots-of-unity condition while avoiding the cyclotomic-subextension obstruction to Malle-type statistics. Its status is open.