Looks like I found a counterexample to a theorem assuming Lang's conjecture, but not sure it is correct.

Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture p. 2

**Theorem 1.3**. Let $K$ be a finitely generated field over $\mathbf{Q}$.
Let $n \ge 8$ be an integer and let
$\alpha_i, (i=0,\ldots,n)$ be fixed elements of $K$.
Suppose Conjecture 1.2 holds for $k$ a finitely
generated field over $\mathbf{Q}$ (cf. [2]).
Then there are only finitely many elliptic curves of the
form $y^2=a x^4 + b x^2 +c \; (a,b,c \in K)$ which have
$\alpha_i$ as the $x$-coordinates of some
$K$-rational points.
In particular the Mordell–Weil ranks of such elliptic curves are
bounded.

Let $K=\mathbb{Q}[\sqrt{21}], a=c=\frac{-\frac{112}{75} s^{4} + \frac{6647}{4725} s^{2} - \frac{83521}{33339600}}{s^{2}} ,b=\frac{\frac{1799}{75} s^{4} - \frac{171377}{37800} s^{2} + \frac{21464897}{533433600}}{s^{2}}, s \in K$.

Let $P(x)=a x^4 + b x^2 + a$. The discriminant depends on $s$ and $P(x)=P(-x)$.

Consider the elliptic curve $y^2=P(x)$ for $s$ for which the discriminant doesn't vanish.

Let $n=9$ and $\alpha_i=\{1,2,4,1/2,1/4,-1,-2,-4,-1/2,-1/4\}$

$$P(1)= \left(21\right) \cdot s^{-2} \cdot (s - \frac{17}{84})^{2} \cdot (s + \frac{17}{84})^{2}$$ $$P(2)= \left(\frac{1764}{25}\right) \cdot s^{-2} \cdot (s^{2} + \frac{289}{7056})^{2}$$ $$P(4)= 289$$ $$P(1/2)= \left(\frac{441}{100}\right) \cdot s^{-2} \cdot (s^{2} + \frac{289}{7056})^{2}$$ $$P(1/4)= 289/256$$

All of the above are squares as are $P(-x)$.

For all infinitely many admissible choices of $s$, $\alpha_i$ are $x$-coordinates.

The $j$ invariant of the Jacobian depends on $s$.

Is this really a counterexample to

Theorem 1.3?

$a,b$ in machine readable form:

```
a=c=-1/33339600*(289+7056*s^2)^2/s^2+289/189
b=257/533433600*(289+7056*s^2)^2/s^2-4913/756
```