Bourgain–Gamburd–Sarnak conjecture on almost-prime values on algebraic groups

Let $G\subset GL_n(\mathbb{R})$ be the group of real points of an algebraically connected, algebraically simply connected, absolutely almost simple linear algebraic group defined over $\mathbb{Q}$. Let $\mathcal{O}$ be a Zariski-dense subgroup of $G(\mathbb{Z})=G\cap GL_n(\mathbb{Z})$, and let $f$ be a nonzero polynomial in $\mathbb{Q}[G]$ that is not a unit, assumes integral values on $\mathcal{O}$, and factors into $k$ irreducibles in $\mathbb{Q}[G]$. Define $r_0(\mathcal{O},f)$ to be the least integer $r$ such that the set of $x\in\mathcal{O}$ for which $f(x)$ has at most $r$ prime factors is Zariski-dense in $G$, the Zariski closure of $\mathcal{O}$. Call $(\mathcal{O},f)$ primitive if, for every integer $q\geq 2$, there exists $x\in\mathcal{O}$ such that $\gcd(f(x),q)=1$. Bourgain–Gamburd–Sarnak conjecture. For every primitive pair $(\mathcal{O},f)$ as above, one has

$$r_0(\mathcal{O},f)=k.$$

This predicts that the number of prime factors achievable densely in the orbit equals the number of irreducible factors of the polynomial. The conjecture is attributed in the source to Bourgain, Gamburd and Sarnak; the supplied material gives no resolution status.