Polynomial correction conjecture for degree growth

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map, and let $\delta_\varphi=\lim_{n\to\infty}(\deg(\varphi^n))^{1/n}$ be its dynamical degree. Define

$$\ell_\varphi=\inf\left\{\ell\geq0:\sup_{n\geq1}\frac{\deg(\varphi^n)}{n^\ell\delta_\varphi^n}<\infty\right\}.$$

Polynomial correction conjecture. The infimum $\ell_\varphi$ exists and is an integer satisfying $0\leq\ell_\varphi\leq N$.

The conjecture asserts polynomially controlled deviation of degree growth from $\delta_\varphi^n$, with an integral exponent bounded by the dimension. A stronger expectation mentioned in the paper is that $\deg(\varphi^n)\asymp n^{\ell_\varphi}\delta_\varphi^n$, but that stronger statement is not part of the conjecture recorded here.