Pivot-minor Erdős–Hajnal conjecture

Let $H$ be a graph. For a graph $G$, let $\boldsymbol{\alpha}(G)$ be its maximum independent-set size and $\boldsymbol{\omega}(G)$ its maximum clique size. Pivot-minor Erdős–Hajnal conjecture. For every graph $H$, there exists $\varepsilon>0$ such that every graph $G$ with no pivot-minor isomorphic to $H$ satisfies

$$\max(\alpha(G),\omega(G))\ge \lvert V(G)\rvert^\varepsilon.$$

This is the proposed analogue of the Erdős–Hajnal conjecture for pivot-minor containment and is stronger than the corresponding vertex-minor theorem. Its status is not explicitly identified in the supplied passage.