Curve-counting conjecture for rationality of $X_{4,6}$

Let $E$ be the elliptic curve used in the construction, and let $H_{144}$ be the matrix defining the curves under consideration. For $p\in E^4$, denote by $\#(p)$ the number of curves $\tilde{C}$ of this type corresponding to $H_{144}$ and containing $p$. Define

$$D_{0,1,2,1}=\{(p_1,p_2,p_3,p_4)\in E^4\mid p_2+2p_3+p_4=0\}.$$

Curve-counting conjecture for $X_{4,6}$. For a generic point $p\in E^4$, one has $\#(p)=1$. Moreover, for a generic point $p\in D_{0,1,2,1}$, one has $\#(p)=1$. This conjecture implies the rationality of $X_{4,6}$ by the cited lemma. It strengthens the curve-counting statement needed beyond the already established unirationality of $X_{4,6}$; the conjecture remains unresolved in the supplied source.