Uniform boundedness of rational points on hyperbolic curves

Let $X$ be a hyperbolic curve defined over a number field $k$, and let $g$ be its genus. Uniform rational-point bound conjecture. The number of rational points of $X$ is bounded above by a bound depending only on $g$ and $k$. The source calls this a stronger folklore version of Mordell's conjecture and gives no resolution of the uniform bound stated here.