The Cremona equivalence conjecture for almost-grid-free hypersurfaces

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$. A set $V\subset\mathbb{P}^s\times\mathbb{P}^s$ is almost-$(s,t)$-grid-free if there are nonempty Zariski-open sets $X,Y\subset\mathbb{P}^s$ such that $V\cap(X\times Y)$ is $(s,t)$-grid-free. Two sets are almost equal if there exist nonempty Zariski-open sets $X,Y\subset\mathbb{P}^s$ on which their intersections agree. The Cremona group is $\operatorname{Cr}(\mathbb{P}^s)$. If $H$ is almost-$(s,t)$-grid-free, then there exist $\sigma\in\operatorname{Cr}(\mathbb{P}^s)$ 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 $H$ is almost equal to $\{F\circ(\operatorname{id}\times\sigma)=0\}$. This is proposed as the obstruction-free replacement for the false statement that every almost-grid-free hypersurface itself has bounded degree after a suitable equivalence; the irreducibility hypothesis is essential, and the conjecture is not resolved in general.