According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.

Let $f$ be univariate squarefree polynomial with integer coefficients
of degree $n$.

Consider the surface:

$$ y^2=f(x)f(z) \qquad (1)$$

According to [MSE comment](https://math.stackexchange.com/questions/1410192/is-there-affine-surface-of-general-type-of-the-form-y2-fx-fz-or-y2-fx#comment2873500_1410192) (1) is of general type for $n \ge 6$ in general.

For rational $d$ consider the twist of the hyperelliptic curve:

$$ dy^2=f(x) \qquad (2)$$

If (2) has two rational points $(x_1,y_1),(x_2,y_2)$ infinitely often, 
this gives
rational point on (1) and on the curve $z=x_2,y^2=f(x)f(z)$.

Since the number of curves must be finite, this means there must
be finite set of relations between $x_1$ and $x_2$, the simplest beeing
$x_1=x_2$ and similar automorphisms.

>Q1 Does Bombieri-Lang imply such restriction on rational points
on the twist?

>Q2 Is there reason to believe there will be simple relation
between $x_1$ and $x_2$ infinitely often?


Assuming $abc$,$abcd$ and significant restrictions on $f$
and certain other non-vanishing conditions,$n$ sufficiently large, 
Granville showed there are more than one non-trivial solutions
to (2) finite number of times.