The classification conjecture for APN power functions with exponent $2^t-1$

Let ${\mathbb F}_{2^n}$ be the finite field with $2^n$ elements, and for $2\leq t\leq n-1$ define

$$G_t:{\mathbb F}_{2^n}\to {\mathbb F}_{2^n},\qquad G_t(x)=x^{2^t-1}.$$

A function is APN if its differential uniformity is at most $2$. The classification conjecture. If $G_t$ is APN, then either $t=2$, or $n$ is odd and

$$t\in\left\{\frac{n+1}{2},n-1\right\}.$$

The three listed cases are respectively the quadratic function $x\mapsto x^3$, the function with $t=(n+1)/2$ for odd $n$, and the inverse permutation with $t=n-1$ for odd $n$. The conjecture asks whether these are the only APN functions in this family; the supplied source gives no resolution status.