Budaghyan et al.'s differential spectrum conjecture for a power function over finite fields

Let $q=2^n$, let $d=q^3+q^2+q-1$, and consider the power function $F(x)=x^d$ over $\mathbb{F}_{q^4}$. For $b\in\mathbb{F}_{q^4}$, consider the equation

$$x^d+(x+1)^d=b.$$

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

This conjecture specifies the differential spectrum of this family of power functions, a topic motivated by the study of resistance to differential cryptanalysis. It was proposed from computational data; its resolution is not indicated in the supplied text.