The relative sizes of coordinates of a point on projective genus 1 curve (second try)

Question

Hopefully this is better than what I asked yesterday and Milton solved.

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 > 1$

$$ \frac{\log \max(|X|,|Y|,|Z|)}{\log \min(|X|,|Y|,|Z|)} > a \qquad (1) $$ finitely often?

Degrees $3,4$ are of interested too.

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I believe Granville-Langevin conjecture (and $abc$) imply some special cases. $ x^3 + k y^3 = m z^3$ with coprime $x,y$ is just on the border of GL.

If (1) suitably fails for this curve GL fails for $ x y (x^3 + k y^3)$.

Answer 1

Let $f$ be a nonconstant rational function on your curve $C$. For any point $P\in E(\mathbb{Q})$, write $$ f(P) = \frac{a_f(P)}{b_f(P)} \in \mathbb{Q} $$ in lowest terms. Then Siegel's theorem implies that $$ \lim_{P\in E(\mathbb{Q}), h(P)\to\infty} \frac{\log|a_f(P)|}{\log|b_f(P)|} = 1. $$ This looks pretty close to what you want, if you take (say) $f$ to be first $X/Y$, then $Y/X$, then $X/Z$, etc.