According to Hall–Lang conjecture there are absolute constants $C$ and $κ$ such that for every elliptic curve $E/Q$ given by a Weierstrass equation $E:=y^2=x^3+ax^2+b$ with $a, b \in \mathbb{Z}$ and for every integral point $P ∈ E(Q)$, i.e., satisfying $x(P) ∈ \mathbb{Z}$, we have-
$$x(P)\leq C \times \text{max}\{|a|, |b|\}^κ$$
Here, $x(P)$ is the coordinate on $x$ axis of the point $P$ on the ellptic curve $E$. The above conjecture gives an upper bound of $x(P)$. Is there anything known regarding the lower bound of $x(P)$ in terms of $a, b$ ?