Uniform dynamical-degree lower-bound conjecture for dominant rational maps

Let $N\geq 1$ and let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map. Its first dynamical degree is

$$\delta(f)=\lim_{n\to\infty}\bigl(\deg(f^n)\bigr)^{1/n}.$$

Uniform dynamical-degree lower-bound conjecture. There exists a constant $\gamma_N>0$ such that, for all dominant rational maps $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$,

$$\delta(f)\geq\gamma_N\cdot\min_{0\leq k<N}\frac{\deg(f^{k+1})}{\deg(f^k)}.$$

For $N=2$, an analogous inequality for birational maps is known with an absolute positive constant, but the stated generalization to all dimensions and dominant rational maps is posed as an open question.