Toral Affine Sieve Conjecture for hyperbolic toral orbits

Let $\g\in\operatorname{GL}_2(\mathbb{Q})$ be a hyperbolic matrix, that is, one having two distinct real eigenvalues; equivalently

$$\operatorname{tr}(\g)^2-4\det(\g)>0.$$

Let $\G=\langle\g\rangle^+:=\{\g^n:n\ge0\}$ be the semigroup generated by $\g$, and suppose that $\bv_0\in\mathbb{Q}^2\setminus\{(0,0)\}$ is a nonzero vector such that the orbit $\cO:=\G\cdot\bv_0\subset\mathbb{Z}^2$ is integral and infinite. Toral Affine Sieve Conjecture. Then

$$\liminf_{(x,y)\in\cO}\frac{\gW(xy)}{\log\log|xy|}\ge \gb_2\approx0.373365.$$

The conjecture predicts a lower bound for the number of prime factors along an infinite toral orbit, in the spirit of the affine sieve. Its status is not resolved in the supplied source context.