The strong existential closedness conjecture for the modular j-function

Let $n\geq 1$ and let $V\subseteq\mathbb{C}^n\times\mathrm{Y}(1)^n$ be an algebraic variety. A point of $V$ is generic over a field $K$ if its transcendence degree over $K$ equals $\dim V$. The strong existential closedness property (SEC) requires that, for every finitely generated field $K\subset\mathbb{C}$ over which $V$ is defined, $V$ contain a point $(\mathbf{z},j(\mathbf{z}))$ generic over $K$. Strong existential closedness conjecture. For every positive integer $n$, every broad and free algebraic variety $V\subseteq\mathbb{C}^n\times\mathrm{Y}(1)^n$ satisfies (SEC). This strengthens existential closedness from the existence of one intersection point to generic intersection points over every finitely generated field of definition. The supplied text gives no proof of the general assertion.