Aubry–McGuire–Rodier exceptional APN polynomial conjecture

Let $q$ be a power of two, and let $f\in\mathbb{F}_q[X]$ be an exceptional APN polynomial, meaning that $f$ induces an APN function on $\mathbb{F}_{q^r}$ for infinitely many positive integers $r$. The Gold monomials are $X^{2^k+1}$ and the Kasami–Welch monomials are $X^{4^k-2^k+1}$ for positive integers $k$; two polynomials are understood to be equivalent in the sense used for APN polynomials. Aubry–McGuire–Rodier conjecture. If $f\in\mathbb{F}_q[X]$ is an exceptional APN polynomial, then $f$ is equivalent to a Gold or a Kasami–Welch monomial. This conjecture proposes a complete classification of exceptional APN polynomials; the cited context notes that Gold and Kasami–Welch monomials are known examples, while the general classification remains open.