Hypothesis T for finite groups and normal subgroups

Let $G$ be a finite group and let $N\unlhd G$ be a normal subgroup. For an irreducible representation $\rho$ of $G$, write $\ker\rho$ for its kernel; for a subgroup $H\subseteq G$, let $\chi$ be a one-dimensional character of $H$, and let $\operatorname{Ind}_H^G\chi$ denote its induced character. Hypothesis $\mathrm{T}(G,N)$. If $\rho$ is irreducible and $N\not\subseteq\ker\rho$, then there are rational numbers $c_{\rho,\chi}$ such that

$$\operatorname{tr}\rho=\sum_{H\subseteq G}\sum_{\substack{\chi\in\operatorname{Irr}(H)\\ \dim\chi=1\\ H\cap N\not\subseteq\ker\chi}}c_{\rho,\chi}\operatorname{Ind}_H^G\chi.$$

The paper proposes this as a group-theoretic hypothesis that would allow holomorphy and non-vanishing information to be transferred from suitable zeta-function quotients to Artin $L$-functions. Its validity for all finite groups and normal subgroups is not established in the supplied text.