Finiteness conjecture for structurally atypical intersections

Let $X$ be a complex analytic space with a special structure, and let $M\subset X$ be a closed subvariety. A product subvariety $Y_1\times Y_2\subset M$ is maximal if it is not properly contained in a larger product subvariety with compatible product structures. A subvariety $W=W_1\times W_2\subset X$ is a weakly special product subvariety if both $W_1\times\{\mathrm{point}\}$ and $\{\mathrm{point}\}\times W_2$ are weakly special in $X$; an $M$-weakly special subvariety is a weakly special subvariety of $X$ contained in $M$. Write $\langle Y_1\times Y_2\rangle_{\mathrm{ws}}$ for its smallest weakly special subvariety.

Finiteness conjecture. There exists a finite set $\Sigma_{\mathrm{p}}\cup\Sigma_M$, where $\Sigma_{\mathrm{p}}$ consists of weakly special product subvarieties and $\Sigma_M$ consists of $M$-weakly special subvarieties of $X$, such that every maximal product subvariety $Y_1\times Y_2\subset M$ satisfies either

$$\langle Y_1\times Y_2\rangle_{\mathrm{ws}}=W_1\times W_2\in\Sigma_{\mathrm{p}},$$

or $Y_1\times Y_2\subset W\subset M$ for some $W\in\Sigma_M$.

This conjecture is already unknown for abelian varieties; the source also explains that the $\Sigma_M$ alternative is necessary for Shimura varieties.