Canonical-height finiteness and arithmetic-degree conjecture for plane polynomial maps

Let $f:{\mathbf{A}}^2\to{\mathbf{A}}^2$ be a dominant polynomial mapping over $\overline{{\mathbf{Q}}}$ with first dynamical degree $\lambda_1>1$ and second dynamical degree $\lambda_2$. Define the arithmetic degree by

$$\alpha(P):=\limsup_{n\to\infty}(h\circ f^n(P))^{1/n}$$

and the canonical height by

$${\hat{h}}(P):=\limsup_{n\to\infty}\frac{h\circ f^n(P)}{n^l\lambda_1^n},$$

where $l\in\{0,1\}$ is determined by $\deg(f^n)\sim n^l\lambda_1^n$. Canonical-height finiteness and arithmetic-degree conjecture. One has ${\hat{h}}(P)<\infty$ for every $P\in{\mathbf{A}}^2(\overline{{\mathbf{Q}}})$. Moreover, if $\lambda_2<\lambda_1^2$, then

$${\hat{h}}(P)=0\quad\Longrightarrow\quad\alpha(P)<\lambda_1.$$

This conjecture concerns the relationship between canonical height and orbit-height growth for dominant plane polynomial maps. The source states it after recalling Silverman's implication in the opposite direction; no resolution is supplied.