Stoll's uniform polynomial height-bound conjecture for genus-2 curves

Let ${\mathcal C}_N$ be the family of genus-$2$ curves defined by integral binary sextics with coefficients bounded in absolute value by $N$. For a rational point $P$ on such a curve, let $H(P)$ denote the height of its $x$-coordinate. Stoll's uniform height-bound conjecture. There are constants $\kappa$ and $B$ such that every rational point $P$ on every curve $C\in{\mathcal C}_N$ satisfies

$$H(P)\leq BN^\kappa.$$

The conjecture asserts that rational-point heights are polynomially bounded in the coefficient height uniformly across the whole family. For quadratic twists of a fixed curve, the paper notes that the ABC conjecture implies such a bound with $\kappa=1/2$; an unconditional general bound remains open.