Broadness and freeness conjecture for intersections with the Gamma graph

Let $n$ be a positive integer. For coordinate projections $\mathrm{Pr}_{\mathbf{i}}$ onto any selected $\ell$ pairs of coordinates, call an algebraic set $V\subseteq\mathbb{C}^{2n}$ broad if every such projection has dimension at least $\ell$. Let the graph of $\Gamma$ be $\{(\mathbf{x},\Gamma(\mathbf{x})):\mathbf{x}\in\mathbb{C}^n\}$. Broadness and freeness conjecture for the Gamma graph. If $V\subseteq\mathbb{C}^{2n}$ is irreducible, broad, and not contained in any subvariety $X_i+a_1X_j+a_2=0$ with $a_1,a_2\in\mathbb{Q}$, then $V$ intersects the graph of $\Gamma$ in a Zariski-dense subset of $V$. This is the proposed geometric criterion for Zariski-dense intersections with the Gamma graph and is presented without a resolution in the source.