Blanc–Van Santen's Perron conjecture for polynomial maps

Let $K$ be a field of characteristic $0$, let $d$ be a positive integer, and let $f\colon \mathbb{A}^d_K\to\mathbb{A}^d_K$ be a polynomial map. Recall that a real number $\lambda\ge 1$ is a Perron number if it is an algebraic integer and every Galois conjugate $\mu$ satisfies $|\mu|\le\lambda$. Blanc–Van Santen's conjecture. The dynamical degree of $f$ is a Perron number of degree at most $d$. This conjecture concerns the arithmetic restrictions on dynamical degrees of polynomial maps; the preceding theorem establishes an algebraicity and degree bound under a properness and spectral-gap hypothesis, while the conjecture predicts the Perron property and the degree bound without that restriction.