The mixed-subdivision strengthening of the planar discrete Brunn–Minkowski conjecture

Let $A,B\subset\mathbb R^2$ be finite two-dimensional sets. A triangulation $T_A$ of $[A]$ uses $A$ as its vertex set, and a triangulation $T_B$ of $[B]$ uses $B$ as its vertex set. Let $M$ be a corresponding mixed subdivision of $[A+B]$, and let $M_{11}$ denote the set of parallelograms in $M$. Mixed-subdivision strengthening. There exist triangulations $T_A$ and $T_B$ and a corresponding mixed subdivision $M$ such that

$$|M_{11}|\geq \sqrt{|T_A|\cdot |T_B|}.$$

This is stated as a stronger version of the planar discrete Brunn–Minkowski conjecture. The parser marks it disproved: the paper gives an example showing that one cannot in general fix the triangulations $T_A$ and $T_B$ in the conjecture.