Necessity and sufficiency for PcN power functions over finite fields

Let $m$ be a positive integer, let $c\in \operatorname{GF}(2^m)$, and let $F(x)=x^d$ be a power function over $\operatorname{GF}(2^m)$. A function is PcN when its $c$-differential uniformity is one. For a positive integer $k$, write $v_2(k)$ for the exponent of $2$ in $k$, and interpret multiplicative inverses modulo $2^m-1$ in the usual way.

PcN power-function conjecture. $F(x)=x^d$ is a PcN function if and only if one of the following conditions holds:

$$d=2^j\quad\text{for }0\leq j\leq m-1,\qquad c\in \operatorname{GF}(2^m)\backslash\{1\};$$

or $d$ belongs to

$$\left\{2^j(2^k+1):j=0,1,\ldots,m-1\right\}$$

or to the set of their multiplicative inverses modulo $2^m-1$, for some positive integer $k$ satisfying $v_2(m)\leq v_2(k)$, and

$$c\in \operatorname{GF}(2^{\gcd(k,m)})\backslash\{1\}.$$

The sufficiency is established by the preceding corollary, while numerical experiments verify the necessity for $2\leq m\leq 10$. Whether the same characterization holds for every $m$ remains open.