The existential and strong existential closedness conjectures for the J-function

Let $n\geq 1$, let $\mathrm{Y}_2(1)$ be the target variety for the tuple-valued function $J$, and let $\mathrm{E}_J^n$ be its graph over $\mathbb{H}^n$. A variety $V\subseteq\mathbb{C}^n\times\mathrm{Y}_2(1)^n$ is $J$-broad when every projection to $\ell$ indexed coordinate blocks has dimension at least $3\ell$, and it is $J$-free as defined in the paper. The properties $(\mathrm{EC})_J$ and $(\mathrm{SEC})_J$ mean respectively Zariski-dense intersection with $\mathrm{E}_J^n$ and existence of a point of that graph generic over every finitely generated field of definition. J-function existential closedness conjecture. For every positive integer $n$, every $J$-broad and $J$-free algebraic variety $V\subseteq\mathbb{C}^n\times\mathrm{Y}_2(1)^n$ satisfies $(\mathrm{EC})_J$; moreover, such $V$ satisfies $(\mathrm{SEC})_J$. This is the analogous conjectural existential-closedness statement for the tuple-valued $J$-function, with the strong version asserting generic solutions. No resolution is supplied.