Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or perhaps by a number depending just on $q$ but not on the elliptic curve)?

The corresponding question for $\mathbb{C}(T)$ has a positive answer by the work of Hindry and Silverman (Invent. math. 93, 1988), who prove more precisely Lang' conjecture: the height of a non-torsion and non-constant $\mathbb{C}(T)$-rational point is larger than an absolute constant multiple of the degree of the minimal discriminant.

However, their proof uses the ABC conjecture, which does not hold in unaltered form for $\mathbb{F}_q(T)$. Does the argument go through if we restrict only to elliptic curves whose $j$-invariant has non-zero derivative? (Szpiro's discriminant-conductor inequality holds for such curves with the absolute constant $6$.) Is there any reason (involving for example some use of the Frobenius substitution $T \to T^p$ on the parameter producing curves of high Szpiro ratio) that Lang's height conjecture may not hold for $\mathbb{F}_q(T)$ without any restriction?