Dynamical Lang conjecture for rational maps

Fix an embedding of the moduli space $\mathscr{M}_d$ in projective space, with associated height $h_{\mathscr{M}_d}$. Let $K$ be a number field and $d\geq2$. Dynamical Lang conjecture. There is a positive constant $C(K,d)$ such that

$$\widehat h_\phi(P)\geq C(K,d)\max\left\{\log\left(N_{K/\mathbb{Q}}\mathfrak{R}_\phi\right),h_{\mathscr{M}_d}(\langle\phi\rangle)\right\},$$

for all rational maps $\phi\in K(z)$ of degree $d$ and all non-preperiodic points $P\in\mathbb{P}^1_K$. This is a dynamical analogue of Lang's height conjecture and remains unresolved in the supplied text.