The fractal intersection conjecture

Let $X$ be an irreducible variety defined over a finitely generated field $K$, and let $F\subset X(\bar K)$ be a fractal on $X$. For a reduced subscheme $Z$ of $X$ defined over $K$, write $\overline{Z(\bar K)\cap F}^{\,\mathrm{Zar}}$ for the Zariski closure of its intersection with $F$.

The fractal intersection conjecture. The closure $\overline{Z(\bar K)\cap F}^{\,\mathrm{Zar}}$ is a union of finitely many points and finitely many components $B_i$ such that $B_i(\bar K)\cap F$ is a fractal in $B_i$ for each $i$.

This conjecture provides a common framework for the Manin–Mumford and Lang-type finiteness results for fractals. The source describes generalized Lang-type statements and several special cases as evidence, but gives no resolution of the conjecture itself.