Cebatorev-set power-residue density conjecture

Let $\rho$ be a motivic Galois representation with open image as above, let $m\geq 2$, and let $\mathcal{P}$ be a Cebatorev set, meaning that, up to finitely many primes, it is specified by a conjugacy-stable subset of the Galois group of a finite Galois extension of $\mathbf{Q}$. Define $\delta_m(\rho;\mathcal{P})$ by restricting both densities in $\delta_m(\rho)$ to $\mathcal{P}$. Cebatorev-set power-residue conjecture. For any Cebatorev set $\mathcal{P}$,

$$\delta_m(\rho;\mathcal{P})=\frac{1}{m}.$$

This is presented as a stronger form of the preceding conjecture and would produce positive-density prime sets not describable in terms of Cebatorev sets; no resolution is stated.