Polynomial hyperbolicity and products of free groups

In this article, we define a locally finite graph $X$ as $\eta$-polynomially hyperbolic if there exists a Lipschitz map $\phi :X\to Z$ to some hyperbolic space $Z$ satisfying the following condition: there exists $C\ge 0$ such that $$\mathrm{\mid }B(p,R_1)\cap {\phi }^{-1}(B(q,R_2))\mathrm{\mid }\le (C{R}_1{)}^{\eta (C{R}_2)}\text{ for all }p,q\in X,R_1,R_2\ge 0.$$ The picture to keep in mind is that coarse fibres of $\phi$ have polynomial growth with a degree coarsely controlled by $\eta$ as the thickness of the fibres grows. The map $\eta$ quantifies how brutal we have to be in order to turn $X$ into a hyperbolic space. Our main result is that, among cocompact special groups, being $\mathrm{l}\mathrm{i}\mathrm{n}$-polynomially hyperbolic amounts not to contain ${\mathbb{F}}_2\times {\mathbb{F}}_2$ as a subgroup. Consequently, containing ${\mathbb{F}}_2\times {\mathbb{F}}_2$ as a subgroup turns out to be quasi-isometric invariant for cocompact special groups.