Curve-counting conjecture for rationality of $X_{5,6}$

Let $E$ be the elliptic curve used in the construction, let $H_{13824}$ be the matrix defining the curves under consideration, and let $\#(p)$ be the curve count defined for points of $E^5$. Define

$$D_{1,0,\zeta,0,-1}=\{(p_1,p_2,p_3,p_4,p_5)\in E^5\mid p_1+\zeta p_3=p_5\}.$$

Curve-counting conjecture for rationality of $X_{5,6}$. For a generic point $p\in D_{1,0,\zeta,0,-1}$, one has $\#(p)=1$. Together with rationality of $X_{4,6}$, the source states that this conjecture implies rationality of $X_{5,6}$; no resolution is supplied.