Hall's conjecture on integral points of Mordell curves

Let $\epsilon>0$, let $B$ be an integer, and let $E_B$ be the Mordell curve

$$y^2=x^3+B.$$

For an integral point $P$ on $E_B$, write $x(P)$ for its $x$-coordinate.

Hall's conjecture. Given $\epsilon>0$, there exists a positive constant $C_\epsilon$ such that, for every integral point $P$ on $E_B$,

$$\left\lvert x(P) \right\rvert < C_\epsilon |B|^{2+\epsilon}.$$

This conjecture gives a uniform upper bound for the $x$-coordinates of integral points in the Mordell-curve family. The source describes it as a well-known conjecture and uses it to compare with lower bounds for heights, but does not report a resolution.