The rational density conjecture for iterated line intersections

Let $S$ be a finite set of points in the plane with rational coordinates. Write $\mathcal{T}(S)$ for the set of points that are intersections of two lines, each determined by a pair of distinct points of $S$, and say that $S$ has finite order if $\mathcal{T}^{i}(S)=\mathcal{T}^{i+1}(S)$ for some $i\geq 0$. Rational density conjecture. If and only if $S$ has infinite order, then

$$\bigcup\nolimits_{i \geq 0}{\mathcal{T}^{i} \left( S \right)} = \mathbb Q^2.$$

This conjecture asks for the exact analogue over the rational plane of the density theorem for finite point-sets: infinite order is expected to produce every rational point, rather than merely a dense subset of the real plane. The source describes this as a conjectural result; no resolution is supplied here.