Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the symmetric group $S_k$ on $G^k$.

Let $A=A(L,D)$ be the set of all integers $n$ which are products of $k$ primes $n=p_1\dots p_k$ (variant: $k$ distinct primes) such that $(Frob_{p_1},\dots,Frob_{p_k}) \in D.$

(Here $Frob_p$ is a Frobenius at $p$ in $G$, assuming $p$ is unramified in $L$. if one of the $p_i$ is ramified, by convention say that $n$ is not in $A$.)

Let $A(x)$ be the number of integers $n<x$ in $A$.

Is there a known equivalent for $A(x)$ when $x$ goes to infinity?

Note that the case $k=1$ is Chebotarev's density theorem. In this case, $A(x) \sim \frac{|D|}{|G|} x/\log x $. In the general case, I would expect $A(X) \sim c x/\log x \frac{(\log \log x)^{k-1}}{(k-1)!}$, with a constant $c$ between $0$ and $1$ depending of $D$, $G$ (and perhaps only of $|D|$, $|G|$), based on the case $k=1$ and the case $D=G^k$, where $A$ is the set of all $k$-almost prime and the desired equivalence is a theorem of Landau. Because of these illustrious limit cases, I would not be surprised if this question was dealt with somewhere in the literature. But I couldn't find where.

I would already be interested in the case where $G$ is abelian. In the case the question takes a more classical form, which I explicit in case some readers are uncomfortable with Chebotarev. We can assume that $G=(\mathbb Z/a\mathbb Z)^\ast$. Then $D$ is a subset of $G^k$ invariant by permutation of the coordinates. The set $A$ is then they of integers $n=p_1\dots p_k$ such that $(p_1 \pmod{a},\dots,p_k \pmod{a})$ in $D$, and the question is still to find an equivalent for the number $A(x)$ of such integers $n$ which are $\leq x$.