Budaghyan et al.'s solution-count conjecture for a finite-field equation

Let $n$ be a positive integer, let $q=2^n$, and set $d=q^3+q^2+q-1$. For $b\in\mathbb F_{q^4}$, consider the equation

$$X^d+(X+1)^d=b$$

in $\mathbb F_{q^4}$.

Budaghyan et al.'s conjecture. The equation has $q^2$ solutions for one value of $b$, $q^2-q$ solutions for $2^n$ values of $b$, and at most $2$ solutions for all remaining points $b$.

This conjecture concerns the differential spectrum of a power function over a finite field and its relevance to almost perfect nonlinear functions in symmetric cryptography. The paper states that the conjecture is proved by the authors' results, while Li et al. had already given an affirmative answer; it is therefore resolved.