Strong pivot-minor Erdős–Hajnal conjecture

Let $H$ be a graph. A set of vertices $A$ is complete to a disjoint set $B$ if every vertex of $A$ is adjacent to every vertex of $B$, and anti-complete if no such edges exist. Strong pivot-minor Erdős–Hajnal conjecture. For every graph $H$, there exists $\varepsilon>0$ such that, for every $n>1$, every $n$-vertex graph with no pivot-minor isomorphic to $H$ has disjoint vertex sets $A$ and $B$ satisfying

$$\lvert A\rvert,\lvert B\rvert\ge \varepsilon n$$

and $A$ is complete or anti-complete to $B$. This stronger property implies the pivot-minor Erdős–Hajnal conjecture, but the supplied text does not give a general resolution.