Morton–Silverman lower-bound conjecture for canonical heights

Let $d\geq 2$, let $\phi\in\mathbb{Q}[z]$ be a polynomial of degree $d$, and let $x\in\mathbb{Q}$. The canonical height $\hat{h}_{\phi}:\mathbb{Q}\to[0,\infty)$ satisfies $\hat{h}_{\phi}(\phi(x))=d\hat{h}_{\phi}(x)$ and vanishes exactly at preperiodic points. Let $h(\phi)$ denote the arithmetic height of $\phi$ in the moduli space of degree-$d$ polynomials.

Morton–Silverman canonical-height conjecture. There is a positive constant $M'=M'(d)>0$ such that, for every polynomial $\phi\in\mathbb{Q}[z]$ of degree $d$ and every point $x\in\mathbb{Q}$ that is not preperiodic for $\phi$,

$$\hat{h}_{\phi}(x)\geq M' h(\phi).$$

This conjecture asks for a uniform lower bound on the canonical height of non-preperiodic rational points relative to the height of the polynomial. It is presented as an analogue of lower bounds for elliptic-curve heights; the source cites a more general version, and the conjecture remains open here.