The broad and Hodge-generic conjecture for quotient maps of Shimura domains

Let $q:

\Omega\to S$be the quotient map associated with the Shimura setting, and let$E_q\subset\Omega\times S$denote its graph. Let$V\subset\mathbb C^{N}\times S$be an irreducible algebraic variety that is broad and Hodge-generic. **The broad and Hodge-generic conjecture.** The intersection$V\cap E_q$is Zariski dense in$V$. This is the conjectural answer to the existential closedness problem for$q$, generalizing analogous conjectures for the exponential and$j$ functions; the source indicates that it is motivated by Ax--Schanuel results for Shimura varieties.