Zilber–Pink finiteness conjecture for optimal subvarieties

Let $S$ be a Shimura variety and let $V$ be a subvariety of $S$. For an irreducible subvariety $W$ of $S$, let $\langle W\rangle$ be the smallest special subvariety of $S$ containing $W$, and define its defect by

$$\delta(W)=\dim\langle W\rangle-\dim W.$$

Call an irreducible subvariety $W$ of $V$ optimal in $V$ if, whenever $W\subsetneq Y$ for another irreducible subvariety $Y$ of $V$, one has $\delta(W)<\delta(Y)$; write $\operatorname{Opt}(V)$ for the set of optimal subvarieties of $V$. Zilber–Pink conjecture. The set $\operatorname{Opt}(V)$ is finite. This is an equivalent formulation of the central unlikely-intersections problem for Shimura varieties, and the paper studies effective results for related weakly optimal loci.