The PcN power-exponent conjecture over seventh-degree finite fields

Let $p$ be an odd prime. For $c=-1$, a power function $x^d$ over $\mathbb{F}_{p^7}$ is perfectly $c$-nonlinear (PcN) when its $c$-differential uniformity is $1$.

PcN power-exponent conjecture. For all $0\leq j\leq 6$, the only possible values of $d$ are the elements of

p^j\left\{1,\frac{p^2+1}{2},(p-1)p^6+p^5+(p-2)p^3+(p-1)p^2+p,\frac{p^4+1}{2},\frac{p^6+1}{2}, (p-2)p^6+(p-2)p^5+(p-1)p^4+p^3+p^2+p,\frac{p^7+1}{p+1}\right\}.

Equivalently, these are the only values of $d$ for which $x^d$ is PcN over $\mathbb{F}_{p^7}$.

The conjecture is based on exhaustive searches for $p=3,5,7$ and seeks to classify all PcN power functions with $c=-1$ over seventh-degree extensions; the stated classification remains unproved in general.