The F1 classification conjecture for Niho type locally-APN power functions

Let $n=2m$. For integers $r<m$ with $\gcd(r,m)=1$, define $s$ by

$$s=\begin{cases}2^r(2^r-1)^{-1},&\text{if }\gcd(2^r-1,2^m+1)=1,\\[4pt]2^r(2^r+1)^{-1},&\text{if }\gcd(2^r+1,2^m+1)=1.\end{cases}$$

Set $F_1(x)=x^{s(2^m-1)+1}$ over $\mathbb{F}_{2^{2m}}$. A power function is Niho type locally-APN if it is both of Niho type and locally-APN. F1 classification conjecture. The function $F_1(x)=x^{s(2^m-1)+1}$ covers all Niho type locally-APN power functions over $\mathbb{F}_{2^{2m}}$. The conjecture is presented as equivalent to a classification of all Niho type locally-APN power functions and remains unresolved in the supplied text.