The affine automorphism conjecture for grid-free hypersurfaces

Let $H$ be an irreducible affine hypersurface in $\mathbb{A}^s\times\mathbb{A}^s$. It is $(s,t)$-grid-free when the associated bipartite algebraic graph contains no complete bipartite subgraph with $s$ vertices in the first part and $t$ vertices in the second. If $H$ is $(s,t)$-grid-free, then there exist $\sigma\in\operatorname{Aut}(\mathbb{A}^s)$ and $F(\overline{x},\overline{y})\in\mathbb{C}[\overline{x},\overline{y}]$ of degree at most $d$ in $\overline{y}$, for some $d=d(s,t)$, such that

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

This is the affine special case of the proposed bounded-degree equivalence principle, with polynomial automorphisms replacing Cremona transformations; the paper presents it as a conjecture and does not resolve it in general.