The positive-density large-triangle conjecture

Let $A\subset\mathbb{R}^2$ be a set of positive upper density, and let $u,v,w\in\mathbb{R}^2$ be non-collinear points. A configuration congruent to a finite set means an image of that set under a Euclidean isometry. Large-triangle conjecture. There exists $l_0>0$ such that, for every $l\geq l_0$, the set $A$ contains a configuration congruent to

$$\{lu,lv,lw\}.$$

This is the simplest case of the paper's proposed extension of positive-density configuration results from non-degenerate configurations with at most $d$ points to configurations containing $d+1$ points. The source describes it as an obvious question left open by earlier results, and no resolution is supplied.