The affine sieve, developed initially by Bourgain-Gamburd-Sarnak in the paper "Affine linear sieve, expanders, and sum-product" published in Inventiones Mathematicae in 2010, deals generally with the following problem: given a subgroup $\Lambda \subset \text{GL}_n$ and a point $b \in \text{GL}_n$, let $\mathcal{O} = \mathcal{O}_b = b \cdot \Lambda$ be the orbit of $b$ under $\Lambda$. If $f \in \mathbb{Z}[x_{jk}]$ satisfies the condition that for all $q \geq 1$ there exists $x \in \mathcal{O}$ such that $\gcd(f(x), q) = 1$m then we say that $f$ is primitive. Let $r_0(\mathcal{O}, f)$ be the least positive integer $r$ such that the set $\{x \in \mathcal{O} : f(x) \text{ has at most } r \text{ prime factors}\}$ is Zariski dense in the Zariski closure of $\mathcal{O}$. If no such integer exists, then we say $r_0(\mathcal{O}, f) = \infty$.

I want to ask if it is possible to use the affine sieve to prove a result of the following shape: Define $k_0(\mathcal{O}, f)$ to be the least positive integer $k \geq 2$ such that the set $$\displaystyle \{x \in \mathcal{O} : f(x) \text{ is } k\text{-free}\}$$ is Zariski dense in the Zariski closure of $\mathcal{O}$. An integer $n$ is said to be $k$-free if for all primes $p | n$, we have $p^k \nmid n$. The $k=2$ case gives the familiar square-free numbers. For a given matrix $\gamma \in \text{GL}_n$, write $\lVert \gamma \rVert$ for the Hilbert-Schmidt norm of $\gamma$. Can the affine sieve be used to prove a lower bound, or even an asymptotic formula, for the quantity $N_{f, \mathcal{O}}(T) = \# \{x \in \mathcal{O} : f(x) \text{ is } k\text{-free}, \lVert x \rVert \leq T\}$?

Initially I thought an obstruction was that a crucial part of the proof of the main theorem of the affine sieve requires showing that the family of Cayley graphs $\mathcal{G}(\Lambda/\Lambda(q), S)$ forms an expander family (here $S$ is a finite symmetric set which generates $\Lambda$) as $q$ ranges over the square-free numbers. The requirement for $q$ to be square-free seems to be the main obstacle, but the above theorem was proved for the case when $q$ ranges over all positive integers in a 2012 paper by Bourgain and Varju. However, I have not seen this information being applied to improve the main theorem of the affine sieve in my literature review.

Any information would be appreciated.