Algebraic-group characterization from a unit cross ratio

Let $a,b,c,d$ be elements in an algebraic matroid, and let $cr(a,b,c,d)$ denote their cross ratio. Algebraic-group characterization conjecture.

$$cr(a,b,c,d)=1$$

in an algebraic matroid if and only if the representation is equivalent to one derived from an algebraic group. This proposes a characterization of representations arising from algebraic groups through a unit cross ratio; the paper gives no resolution or further hypotheses.