Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.


Write a point $P = (X , Y , Z)$ with the smallest coprime integers
$X,Y,Z$.


Is it true that for every fixed $ a > 0$


$$
\log \max(|X|,|Y|,|Z|)- \log \min(|X|,|Y|,|Z|) > a
$$
finitely often?


I believe it is true for homogenized Weierstrass model,
false for genus $0$.


Limited experiments with cubics and quartics suggest it
might be true.


Solution for degrees $3,4$ would be of interest too.



The relative sizes of coordinates of a point on projective genus 1 curve
I don't think so. Suppose that $T=(0,0)$ is a rational point on your curve $C$, and suppose that the rational points on $C$ lie dense around $T$ in the real topology. (It is easy to find such a $C$.) Your assertion would preclude $x=X/Z$ and $y=Y/Z$ from getting arbitrarily close to $(0,0)$, but this is exactly what happens for $x(P)$ and $y(P)$ if $P$ tends to $T$.


