The special-point non-density conjecture for Shimura quotient graphs

Let $\Gamma\subset G(\mathbb Q)$ be an arithmetic subgroup, let $S=\Gamma\backslash\Omega$, and let $V\subset\mathbb C^{N}\times S$ be an irreducible algebraic variety. Let $\pi:V\to\mathbb C^{N}$ denote projection onto the first factor, and assume that $\pi(V)$ is Zariski dense. Let $q:\Omega\to S$ denote the quotient map. A point $x\in\Omega$ is special when its Mumford--Tate group is a torus. The special-point non-density conjecture. The subset of $V$ consisting of points $(x,q(x))$ for which $x$ is special is not Zariski dense in $V$. This question is motivated by the Andr\e--Oort conjecture and asks whether the special points among the points supplied by the existential closedness problem remain non-dense. Its resolution status is not specified in the source.