Lang–Silverman conjecture for Drinfeld modules

Let $K$ be the global function field in the paper's setting. For every finite extension $L/K$, every integer $r\geq 1$, every Drinfeld module $\phi/L$ of rank $r$, and every non-torsion point $x\in\phi(L)$, write $\hat{h}_\phi(x)$ for the canonical height, $h(j_\phi)$ for the height of its $j$-invariant, and $\deg(\mathscr{D}_{\phi/L})$ for the degree of its discriminant divisor. Lang–Silverman conjecture. For every finite extension $L/K$ and every $r\geq 1$, there is an $\varepsilon>0$ such that if $\phi/L$ is a Drinfeld module of rank $r$ and $x\in\phi(L)$ is non-torsion, then

$$\hat{h}_\phi(x)\geq \varepsilon\max\{h(j_\phi),\deg(\mathscr{D}_{\phi/L})\}.$$

This is a conjectural lower bound for canonical heights that strengthens bounds depending only on the number of places of bad reduction. The paper motivates it by analogy with conjectures of Lang and Silverman for elliptic curves; its status is not resolved in the supplied context.