Maximal differential uniformity distribution conjecture for polynomials over finite fields

Let $n$ be a positive integer, set $q=2^n$, and let $f\in\mathbb{F}_{2^n}[x]$ be a polynomial of degree $m>4$. Define its differential uniformity by

$$\delta(f):=\max_{(\alpha,\beta)\in\mathbb{F}_{2^n}^{\ast}\times\mathbb{F}_{2^n}}\sharp\{x\in\mathbb{F}_{2^n}\mid f(x+\alpha)+f(x)=\beta\}.$$

Maximal differential uniformity distribution conjecture. There exists $\varepsilon_m>0$ such that, for all sufficiently large $n$, at least $\varepsilon_m2^{2n}$ pairs $(\alpha,\beta)\in\mathbb{F}_{2^n}^{\ast}\times\mathbb{F}_{2^n}$ satisfy

$$\sharp\{x\in\mathbb{F}_{2^n}\mid f(x+\alpha)+f(x)=\beta\}=\delta(f).$$

The conjecture asserts that a positive proportion, depending only on the degree, of differential inputs attain the polynomial's maximal differential uniformity. The supplied evidence marks this statement as resolved; the surrounding discussion relates it to earlier results showing that maximal differential uniformity is typical for suitable degrees.