The biregular equivalence conjecture for grid-free hypersurface restrictions

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$, and let $X,Y$ be nonempty Zariski-open subsets of $\mathbb{P}^s$. A hypersurface restriction is $(s,t)$-grid-free when it contains no complete bipartite subgraph with $s$ vertices in $X$ and $t$ vertices in $Y$. If $H\cap(X\times Y)$ is $(s,t)$-grid-free, then there exist a subset $Y'\subset\mathbb{P}^s$, a biregular map $\sigma\colon Y\to Y'$, and $F(\overline{x},\overline{y})\in\mathbb{C}_{\mathrm{hom}}[\overline{x},\overline{y}]$ of degree at most $d$ in $\overline{y}$, for some $d=d(s,t)$, such that

$$H\cap(X\times Y)=\{F\circ(\operatorname{id}\times\sigma)=0\}\cap(X\times Y).$$

This is the stronger form of the Cremona conjecture and is proved when $s=1$, and when $s=t=2$ with $Y=\mathbb{P}^2$; it remains open in general.