The uniform exceptional-linear-subspace conjecture in dynamical Mordell–Lang

Let $\f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$, let $P\in{\mathbb P}^n({\mathbb C})$ have Zariski-dense forward orbit

$$\mathcal O_\f(P)=\{\f^n(P):n\ge0\},$$

and let $r\ge1$. An $(r-1)$-dimensional linear subspace $L\subset{\mathbb P}^n$ is super-spanned by $r+1$ points if every subset of $r$ of them spans $L$. Uniform exceptional-linear-subspace conjecture. There are only finitely many such $L$ for which $L\cap\mathcal O_\f(P)$ contains $r+1$ points super-spanning $L$, and their number is bounded by a function depending only on $n$ and $d$. The conjecture concerns exceptional intersections of dense orbits with linear spaces; the paper proves special cases, while the stated general uniform bound remains open.