Rational-point conjecture for the surfaces $\widetilde{Z}(17,1)$ and $\widetilde{Z}(17,3)$

Let $\widetilde{Z}(17,k)$ be the surface in ${\mathbb A}^4$ with coordinates $(T,x,y,z)$ defined by

$$y^2+(T+1)(T-2)xy+T^3y=x^3-x^2$$

and

$$z^2=F_k(T,x,y),$$

where $F_k$ is the polynomial recorded in the appendix.

Rational-point conjecture. The only ${\mathbb Q}$-points on $\widetilde{Z}(17,1)$ lie above one of the listed curves in $(x,y)$, the curve $T=0$, or one of the points in the indicated table; likewise, the only ${\mathbb Q}$-points on $\widetilde{Z}(17,3)$ lie above one of its listed curves in $(x,y)$, the curve $T=0$, or one of the points in that table.

This conjecture seeks a complete description of the rational points on the two explicit surfaces, which arise in the study of pairs of 17-congruent elliptic curves. Determining all such points remains open.