The Lehmer-type height conjecture for polarized dynamical systems

Let $(X,f,\mathscr L)$ be a polarized dynamical system over a number field $K$, with canonical height $\hat h_{\mathscr L}$, and let $x\in X(\overline{\mathbb Q})$ be a non-preperiodic point. Dynamical Lehmer conjecture. There should exist a real number $c>0$ such that, for every non-preperiodic point $x$,

$$\hat h_{\mathscr L}(x)\geq\frac{c}{[K(x):K]}.$$

This is a dynamical analogue of Lehmer's problem. The source says that weaker estimates are known for abelian dynamical systems, whereas the asserted bound is not known for arbitrary polarized dynamical systems, even in dimension one.