Hrushovski's nilpotent control conjecture for approximate subgroups of special linear groups

Let $A\subseteq \operatorname{SL}_n(\mathbb{C})$ be a $K$-approximate group, and say that a set $B$ $K'$-controls $A$ when the source's approximate-group control relation holds. Hrushovski's conjecture. There is a $K'$-approximate group $B$ which is nilpotent and $K'$-controls $A$, where $K'$ depends only on $K$.

The statement is presented as a consequence expected from Hrushovski's model-theoretic work together with structure theory of algebraic groups. The source does not provide a resolution, so the conjecture remains open.