The Kloosterman-sum identity conjecture for the auxiliary sum $K_m'$

Let $m$ be odd and let $k$ satisfy $\operatorname{gcd}(k,m)=1$. Define

$$f(v)=\frac{(v^{2^k}+1)v^{2^k}}{(v^{2^k}+v)^{2^k+1}},$$

with $f(0)=f(1)=0$, and define

$$K_m'=\sum\limits_{x\in \operatorname{GF}(2^m)^*}(-1)^{\operatorname{Tr}_m(f(v))}.$$

Kloosterman-sum identity conjecture.

$$K_m'=K_m.$$

Here $K_m$ is the Kloosterman sum $\sum_{x\in\operatorname{GF}(2^m)^*}(-1)^{\operatorname{Tr}_m(x+x^{-1})}$. This is the second exponential-sum conjecture on which the proposed settlement of the relevant cross-correlation distribution depends. The source gives no evidence of resolution, so its status is open.