Optimal lifting for the projective action of SL3(?)

Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over finite field $F_q$ there exists $\gamma$ $SL_3(Z)$, coordinates bounded by $q^{1/3+\epsilon}$, whose projection $SL_{3}(F_q)$ sends $x$ $y$. The exponent $1/3$ is optimal result rank generalization of Sarnak's strong approximation theorem for $SL_2(Z)$.